Eur. Phys. J. C (2017) 77:17
DOI 10.1140/epjc/s10052-016-4583-x
Regular Article - Theoretical Physics
Off-diagonal deformations of Kerr metrics and black ellipsoids in
heterotic supergravity
Sergiu I. Vacaru1,2,3,a, Klee Irwin1,b
1 Quantum Gravity Research, 101 S. Topanga Canyon Blvd # 1159, Topanga, CA 90290, USA
2 Project IDEI, University “Al. I. Cuza”, Iasi, Romania
3 Flat 4, Brefney House, Fleet Street, Ashton-under-Lyne, Lancashire OL6 7PG, UK
Received: 9 October 2016 / Accepted: 16 December 2016
© The Author(s) 2017. This article is published with open access at Springerlink.com
Abstract Geometric methods for constructing exact solu-
tions of equations of motion with first order α′ correc-
tions to the heterotic supergravity action implying a non-
trivial Yang–Mills sector and six-dimensional, 6-d, almost-
Kähler internal spaces are studied. In 10-d spacetimes, gen-
eral parametrizations for generic off-diagonal metrics, non-
linear and linear connections, and matter sources, when
the equations of motion decouple in very general forms
are considered. This allows us to construct a variety of
exact solutions when the coefficients of fundamental geo-
metric/physical objects depend on all higher-dimensional
spacetime coordinates via corresponding classes of generat-
ing and integration functions, generalized effective sources
and integration constants. Such generalized solutions are
determined by generic off-diagonal metrics and nonlinear
and/or linear connections; in particular, as configurations
which are warped/compactified to lower dimensions and for
Levi-Civita connections. The corresponding metrics can have
(non-) Killing and/or Lie algebra symmetries and/or describe
(1 + 2)-d and/or (1 + 3)-d domain wall configurations, with
possible warping nearly almost-Kähler manifolds, with grav-
itational and gauge instantons for nonlinear vacuum configu-
rations and effective polarizations of cosmological and inter-
action constants encoding string gravity effects. A series of
examples of exact solutions describing generic off-diagonal
supergravity modifications to black hole/ellipsoid and soli-
tonic configurations are provided and analyzed. We prove
that it is possible to reproduce the Kerr and other type black
solutions in general relativity (with certain types of string
corrections) in the 4-d case and to generalize the solutions
to non-vacuum configurations in (super-) gravity/string the-
ories.
a e-mail: sergiu.vacaru@gmail.com
b e-mail: klee@quantumgravityresearch.org
Contents
1 Introduction
. . . . . . . . . . . . . . . . . . . . . .
2 Heterotic supergravity in nonholonomic variables . . .
2.1 Geometric conventions on nonholonomic 2 +
2 + ··· splitting . . . . . . . . . . . . . . . . . .
2.2 The AFDM for heterotic supergravity
. . . . . .
2.3 Decoupling and integration of nonholonomic
equations of motion . . . . . . . . . . . . . . . .
2.3.1 Ansatz for metrics, N-connections, and
gravitational polarizations
. . . . . . . . .
2.3.2 Ricci d-tensors and N-adapted sources . . .
2.3.3 N-adapted sources and nonholonomically
modified Einstein equations
. . . . . . . .
2.3.4 The ansatz for effective matter fields in
heterotic string gravity . . . . . . . . . . .
2.3.5 Decoupling of nonholonomic equations
of motion and effective . . . . . . . . . . .
2.3.6 Integration of nonholonomic equations of
motion by generating functions and effec-
tive sources . . . . . . . . . . . . . . . . .
2.3.7 A nonlinear symmetry of generating func-
tions and effective sources . . . . . . . . .
2.3.8 The Levi-Civita conditions . . . . . . . . .
2.4 Small N-adapted nonholonomic stationary defor-
mations
. . . . . . . . . . . . . . . . . . . . . .
3 Nonholonomic heterotic string deformations of the
Kerr metric . . . . . . . . . . . . . . . . . . . . . . .
3.1 Preliminaries on the Kerr vacuum solution and
nonholonomic variables . . . . . . . . . . . . . .
3.2 Off-diagonal deformations of 4-d Kerr metrics
by heterotic string sources
. . . . . . . . . . . .
3.2.1 Nonholonomically string induced torsion
for Kerr metrics in the 4-d sector . . . . . .
3.2.2 Small modifications of Kerr metrics and
effective string sources . . . . . . . . . . .
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Eur. Phys. J. C (2017) 77:17
3.3 String induced ellipsoidal 4-d deformations of
the Kerr metric
. . . . . . . . . . . . . . . . . .
3.3.1 Ellipsoidal configurations with string induced
cosmological constant
. . . . . . . . . . .
3.3.2 Ellipsoid Kerr–de Sitter configurations in
R2 and heterotic string gravity . . . . . . .
3.4 Extra-dimensional off-diagonal string modifica-
tions of the Kerr solutions . . . . . . . . . . . . .
3.4.1 6-d deformations with nontrivial cosmo-
logical constant . . . . . . . . . . . . . . .
3.4.2 10-d deformations with NS 3-form and 6-
d almost Kähler internal spaces
. . . . . .
3.4.3 Off-diagonal solutions in standard 10-d
heterotic string coordinates . . . . . . . . .
4 Outlook and concluding remarks . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
The problem of constructing exact solutions (in partic-
ular, with parametric dependence on some deformation
parameters) of equations of motion in ten-dimensional,
10-d, (super-) string and gravity is of great importance.
Recent approaches for solutions of generalized gravita-
tional and matter field equations in modified gravity the-
ories (MGTs, with bi-metric/-connection structure, possi-
ble nontrivial mass terms for graviton, locally anisotropic
effects etc.) including generic off-diagonal solutions in gen-
eral relativity, GR, have been elaborated upon. This involves
phenomenological applications in high energy physics, var-
ious approaches to quantum gravity and attempts to explain
observational data in modern accelerating cosmology. For
review of such subjects, we cite, respectively [1–7], on super-
strings, flux compactifications, D-branes, instantons etc; [8–
11], on geometric methods in quantum gravity; and [12–
30], on MGTs and applications; see also the references
therein.
The supergravity/superstring and MGT gravitational field
equations are formulated as sophisticated systems of non-
linear partial differential equations, PDEs. Various types
of advanced analytic and numeric methods for construct-
ing exact and approximate solutions of such equations have
been explored. For GR, a number of examples of exact and
physically important solutions are summarized in mono-
graphs [31,32]. In generalized (super-) gravity theories, most
of the solutions with a variety of different vacua, such
as in string and brane theories, comes from correspond-
ing choices of the internal manifold. Toroidal compactifi-
cations, special geometric cases as Calabi–Yau and, more
generally but with less supersymmetry, with SU (3) structure
manifolds were considered. The bulk of solutions in vari-
ous super/noncommutative/extra-dimension/modified grav-
ity theories are described by metrics, frames, and connec-
tions with coefficients of fundamental geometric/physical
objects depending on one and/or two coordinates in 4-d to
10-d spacetimes. For well-known classes of solutions, diag-
onalizations of metrics are possible via coordinate trans-
forms and the linear connection structures are mostly of Levi
Civita, LC, and Kähler type. Additional distortions by tor-
sion structures are also considered. Various generalizations
of well-known and physically important exact solutions for
the Schwarzschild, Kerr, Friedman–Lemaître–Robertson–
Walker (FLRW), wormhole spacetimes etc were constructed.
These classes of diagonalizable metrics (the off-diagonal
terms in the Kerr solutions are determined by rotations and
respective frame/coordinate systems) are generated by a cer-
tain ansatz where equations of motion are transformed into
certain systems of nonlinear second order ordinary equations
(ODEs), 2-d solitonic equations etc. These systems of ODEs
have Killing vector symmetries which result in additional
parametric symmetries [33–35] and depend on integration
constants.
A number of physically important solutions with black
hole, wormhole, cosmological, monopole, and instanton
configurations etc. were constructed in different (super-)
string/gravity and MGTs for a diagonalizable ansatz depend-
ing generically on one spacetime coordinate (in certain cases,
with dependencies on two coordinates and with Killing or
other type symmetries). For the majority of such solutions,
the corresponding equations of motion transform with the
corresponding diagonal ansatz for metrics and frame trans-
forms from nonlinear systems of PDE in nonlinear sys-
tems of ODEs. The integrals of such ODEs depend on
certain integration constants which are defined from some
boundary/asymptotic/initial and prescribed symmetry con-
ditions. For many years, physicists and mathematicians con-
centrated their efforts on constructing further generaliza-
tions and applications of “diagonalizable” solutions in string
and (super-) gravity theories because it was more “easy”
to find analytic and numeric solutions of resulting systems
of ODEs. In these approaches, the integration constants
can be related to certain physical constants considering the
Cauchy problem, or by using various assumptions on asymp-
totic/boundary/symmetry conditions.
All versions of (supersymmetric) modified Einstein equa-
tions consist of very sophisticated off-diagonal nonlinear sys-
tems of PDEs. In general form, the main properties of nonlin-
ear/nonholonomic/parametric interactions of such (super-)
gravitational and matter field systems are described by PDEs
and not by approximations to ODEs. To find general classes
of solutions in analytical form, understand their geometric
properties and search for possible physical implications is of
great importance in modern mathematics, physics, and cos-
mology. Imposing only a “simple” diagonalizable ansatz of
higher symmetry (as it is usually considered for generating
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17
new classes of exact solutions in various gravity theories),
we “cut” the possibility to find infinite numbers of classes
of generic off-diagonal solutions determined by generating
and integrations functions depending on 3, 4,... spacetime
coordinates, with various commutative and noncommutative
parameters etc from the very beginning. A typical example
is that of solitonic wave solutions depending on three vari-
ables, which cannot be generated for a very “simple” diagonal
ansatz and factorized dependent on certain generated func-
tions. It is possible that various problems in modern accel-
eration cosmology (with structure formation, dark energy
and dark matter, etc.) could be solved by certain (super-)
string/gravity solutions related to generic off-diagonal non-
linear configurations in GR and it may not be necessary to
radically modify the standard gravity and particle physics
theories.
The problem of constructing generic off-diagonal exact
solutions1 with coefficients of metrics and connections and
other physically important geometric objects depending on
three-, four- and extra-dimensional spacetime coordinates
is much more difficult. For instance, there are a maximum
of six independent components of a metric tensor from ten
components in a 4-d (pseudo-) Riemannian spacetime.2 Any
such ansatz with metrics depending on three–four spacetime
coordinates transforms the Einstein equations into systems
of nonlinear PDEs, which cannot be decoupled and inte-
grated in a general analytic form if the constructions are
performed with respect to local coordinate frames and for
the LC-connection. The condition of zero torsion imposes
various types of contractions between the coefficients of the
linear connection, reference frames and various tensor fields
which do not allow any general decoupling of the correspond-
ing systems of PDEs. To generate solutions with generic
off-diagonal metrics and generalized connections for higher-
dimensional configurations (for instance, in 5d–10-d string
gravity and MGTs) is a technically more difficult task than
in 3d–4d theories.
In a series of publications [35–44], the so-called anholo-
nomic frame deformation method, AFDM, of constructing
exact solutions in commutative and noncommutative (super-)
gravity and geometric flow theories has been explored. By
straightforward analytic computations, it was proven that it
is possible to decouple the gravitational field equations and
generate general classes of solutions in various theories of
gravity with metric and nonlinear, N-, and linear connections
structures. The geometric formalism is based on spacetime
fibrations determined by nonholonomic distributions with
splitting of dimensions, 2 (or 3) +2 + 2 + ··· . In explicit
1 The metric fields corresponding to such solutions cannot be diago-
nalized in a finite or infinite spacetime region via coordinate transforms.
2 Four components from a maximum of ten can be fixed to be zero
using coordinate transforms, which is related to the Bianchi identities.
form, certain classes of N-elongated frames of reference, the
considered formal extensions/embeddings of 4-d spacetimes
into higher-dimensional spacetimes are introduced and nec-
essary types of adapted linear connections are defined. These
connections are called distinguished, d-connections, defined
in forms which preserve the N-connection splitting. In Ein-
stein gravity, a d-connection is considered auxiliary, which in
certain canonical forms can be uniquely defined by the metric
structure following the conditions of metric compatibility and
some other geometric conditions (for instance, that certain
zero values for “pure” horizontal and vertical components
contain nonholonomically induced torsion fields). Surpris-
ingly, such a canonical d-connection allows us to decouple
the equations of motion in general forms and generate vari-
ous classes of exact solutions in generalized/modified string
and gravity theories. Having constructed a class of gener-
alized solutions in explicit form (depending on generating
and integration functions, generalized effective sources and
integration constants), we can constrain the induced torsion
fields to zero and “extract” solutions for LC-configurations
and/or Einstein gravity. It should be emphasized that it is
important to impose the zero-torsion conditions at the end,
i.e. after we have found a class of generalized solutions.
We cannot decouple and solve the corresponding systems
of PDEs in general forms if we use the LC-connection from
the very beginning. Here it should be noted that it is impor-
tant to work with nontrivial torsion configurations in order
to find exact solutions in string gravity and gauge gravity
models.
In this paper, we apply methods in the geometry of non-
holonomic and almost-K ähler manifolds in order to study
heterotic supergravity derived in the low-energy limit of het-
erotic string theory [45–47]. This publication is associated
with another paper [48], where an approach to heterotic
string gravity is formulated in the language of nonholonomic
and almost-Kähler geometry. We cite also [1, Section 4.4]
for a summary of previous results and similar conventions
on warped configurations and modified gravitational equa-
tions.3 The main goal of this work is to develop a geometric
method for integrating in generic off-diagonal forms, and
for generalized connections, the equations of motion of het-
erotic supergravity, up to and including terms of order α′.
As a secondary goal, we shall construct explicit examples of
exact solutions describing nonholonomic deformations of the
Kerr metric. In general, it is possible to formulate conditions
for effective sources and generic off-diagonal when string
gravity may encode/mimic equivalent solutions in massive
gravity and/or modified f (R, T ) gravity. For reviews and
3 Nevertheless, we shall elaborate a different system of notations with
N-connections and auxiliary d-connections which allows us to define
geometric objects on higher order shells of nonholonomically splitted
10-d spacetimes.
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original results related to massive and other types MGTs, we
cite references [20–23,49,50].
In this work, a series of exact and/or small param-
eter depending solutions which for small deformations
mimic rotoid Kerr–de Sitter-like black holes/ellipsoids,
self-consistently embedded into generic off-diagonal back-
grounds of 10-dimensional spacetimes are constructed. Such
backgrounds can be of solitonic/vertex/instanton type. We
study string gravity modifications with respect to nonholo-
nomic frames and via re-definition of generating and integra-
tions functions and coefficients of sources. These modifica-
tions can be analyzed in the framework of Einstein grav-
ity but modeled by effective polarized cosmological con-
stants and off-diagonal terms. Using solutions for heterotic
string gravity, it is possible to mimic physically impor-
tant effects in modified gravity. In a series of associated
papers, we studied the acceleration of the universe, cer-
tain dark energy and dark matter locally anisotropic inter-
actions, effective renormalization of quantum gravity mod-
els [8,24,51] via nonlinear generic off-diagonal interac-
tions on effective Einstein spaces. These constructions can
developed for models elaborated in the framework of string
theory.
The solutions of heterotic supergravity which are con-
structed in this and the associated [48] work describe (1+3)-
dimensional walls endowed with generic, off-diagonal met-
rics, warped to an almost-Kähler 6-d internal space in the
presence of nonholonomically deformed gravitational and
gauge instantons. The generalized instanton contributions are
adapted to a nontrivial nonlinear connection structure deter-
mined by generic off-diagonal interactions which allows us
to solve the Yang–Mills, YM, sector and the corresponding
Bianchi identity at order α′ (related to the gravitational con-
stant in 10-d). Such 10-d solutions preserve two real super-
charges, which correspond to the N = 1/2 supersymmetry.
The almost-Kähler internal 6-d structure can be defined for
various classes of solutions in 10-d gravity if we prescribe an
effective Lagrange type generating function. In this approach
we can work both with real nonholonomic gravitational and
YM instanton configurations to consider deformed SU (3)
structures.
The paper is organized as follows: We begin Sect. 2 with
a summary on formulation of the heterotic supergravity the-
ory in nonholonomic variables which was performed in [48].
This will allow us to derive a general decoupling property of
motion equations in further sections. The geometric formal-
ism on nonlinear and distinguished connections and adapted
metrics to nonholonomic 2+ 2+ 2+··· splitting of higher-
dimension spacetimes is outlined. We develop the AFDM
as a geometric method for constructing exact solutions of
equations of motion in heterotic string theory and related 4d–
10d modified Einstein gravitational equations. The contribu-
tions of gauge-like NS 3-forms, curvature of interior almost-
Kä hler configurations, effective scalar and gauge fields
etc. are encoded into certain effective sources and generic
off-diagonal terms of metrics, with effective N-connection
structure, defining a nontrivial vacuum structure. Such gen-
eralized/modified gravitational equations are formulated in
adapted variables and with a very general off-diagonal ansatz
for the metrics, (non-) linear connections and effective matter
fields, which allows a decoupling of corresponding nonlin-
ear systems of PDEs in very general forms. We show that
using this nonholonomic decoupling property, it is possible
to construct classes of exact solutions depending on vari-
ous sets of generating and integration functions and inte-
grations constants, on all 10-d spacetime coordinates. The
existence of a very important nonlinear symmetry is proven.
This allows a re-definition of the generating functions and
effective sources to other equivalent data with effective cos-
mological constants induced by off-diagonal, or warped, and
effective sources interactions. It is shown how the geomet-
ric constructions can be performed for the “simplest” case
of one Killing symmetry in the 4-d case and then general-
ized to non-Killing configurations and higher dimensions.
The conditions of generating solutions with zero torsion are
analyzed. A self-consistent formalism of constructing small
N-adapted nonholonomic stationary deformations is also
explored.
Section 3 is devoted to a rigorous geometric study of non-
holonomic generic off-diagonal deformations of exact solu-
tions in heterotic string gravity containing the Kerr solution
as a “primary” configuration. We show by using the AFDM
it is possible to generate the Kerr solution as a particular non-
holonomically constrained case and small parametric defor-
mations with a well-defined physical interpretation. Then we
construct solutions with general off-diagonal deformations of
the Kerr metrics in 4d to 10d effective gravity with heterotic
string modifications. We provide examples of (non-Einstein)
metrics with nonholonomically induced torsions and study
small off-diagonal modifications of the Kerr metrics deter-
mined by warped and general almost-Kähler internal space
structures. Separate subsections are devoted to ellipsoidal 4-
d, 6-d, and 10-d deformations of the Kerr metric resulting
in target vacuum rotoid, or Kerr–de Sitter, configurations,
all self-consistently defining exact solutions of equations of
motion in heterotic string theory.
Finally (in Sect. 4), we summarize the paper, provide con-
clusions and speculate on physical meanings of solutions
with generic off-diagonal metrics and generalized connec-
tions constructed using the AFDM for the heterotic string
theory.
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2 Heterotic supergravity in nonholonomic variables
In this section, we outline a nonholonomic geometric
approach to the heterotic supergravity modeled in the low-
energy limit of heterotic string theory as a N = 1 and 10-
d supergravity coupled to super Yang–Mills theory; see for
details [48]. The nonholonomic variables will be parameter-
ized in forms which allow a general decoupling of equations
of motion and generating exact solutions depending, in prin-
ciple, on all spacetime coordinates. Such a higher-dimension
spacetime is modeled as a 10-d manifold M, equipped with
a Lorentzian metricǧ of signature (++−+++++++),
with a time-like third coordinate. The heterotic supergravity
theory is defined by a couple (M,ǧ), an NS 3-formȞ , a
dilaton fieldφ̌ and a gauge connection A∇̌,with gauge group
SO(32) or E8 × E8. In our approach, we elaborate a system
of notations which is adapted correspondingly for applica-
tions of geometric methods of constructing exact solutions
in [40–44]. The notations in [48] and this work are differ-
ent from the “standard” system of notations in string theory
(see, for instance, [1]). A so-called nonholonomic “shell by
shell”, or 2 + 2 + ··· , splitting should be elaborated with
corresponding left shell labels and shell indices. This mini-
mizes the procedure of separating the equations of motion in
certain general forms and for constructing exact solutions. In
“shell” diadic variables, certain important (non) linear sym-
metries are explicitly shown and the type of generating and
integration functions which can be considered are empha-
sized. Using only standard 4-d, 6-d and/or 10-d indices, like
in former superstring and supergravity work, it is not possible
to understand how the AFDM can be applied for generating
off-diagonal solutions in (super-) string/gravity theories.
2.1 Geometric conventions on nonholonomic 2 + 2 + ···
splitting
For geometric spacetime models on a 10-d pseudo-Riem-
annian spacetime M with a time-like coordinate u3 = t
and other coordinates being space-like,4 we consider conven-
tional splitting of dimensions dim M = 4 + 2s = 10; s =
0, 1, 2, 3. The AFDM, allows us to construct exact solutions
with arbitrary signatures of metricsǧ, but our goal is to
consider extra-dimensional space generalizations of the Ein-
stein theory to heterotic supergravity models. In most general
forms, this is possible if we use the formalism of nonlinear
connection splitting for higher-dimensional (super-) spaces
and strings, which was originally considered in (super-)
Lagrange–Finsler theory [53,54]. We shall not consider
Finsler type gravity in this work, but we follow an approach
4 This parameterization of coordinates is convenient for constructing
various classes of stationary solutions with warping on coordinate u4
in a “minimal” form.
re-defined for nonholonomic distributions on (super-) man-
ifolds [40,43,44]. The same geometric technique can be
applied for tangent (super-) bundles or for (super-) mani-
folds enabled with certain classes of nonholonomic distribu-
tions like nonholonomic frames, nonlinear connections etc.
For vector/tangent (super-) bundles, certain x-coordinates are
used on the base (super-) manifold, but y-coordinates are con-
sidered for the typical fiber (super-) vector space. In the case
of fibrations, the (x, y)-coordinates are used for the definition
of a fibered structure.
We consider (abstract, or coordinate) indices and coor-
dinates uαs = (xis , yas ) for an oriented number of two-
dimensional, 2-d, “shells” added to a 4-d spacetime of signa-
ture (++−+). For s = 0, we write uα = (xi , ya) and then
extend “shell by shell” to a local system of 10-d coordinates,
s = 0 : uα0 = (xi0 , ya0) = (xi , ya); s = 1 : uα1
= (xα = uα, ya1) = (xi , ya, ya1);
s = 2 : uα2 = (xα1 = uα1 , ya2) = (xi , ya, ya1 , ya2);
s = 3 : uα3 = (xα2 = uα2 , ya3) = (xi , ya, ya1 , ya2 , ya3),
(1)
The corresponding subsets of indices are labeled in the form:
i =
i0, j =
j0,... = 1, 2; a = a0, b = b0,... =
3, 4, when u3 = y3 = t ; a1, b1 ... = 5, 6; a2, b2 ... =
7, 8; a3, b3 ... = 9, 10; and, for instance, i1, j1,... =
1, 2, 3, 4; i2, j2,... = 1, 2, 3, 4, 5, 6; i3, j3,... = 1, 2, 3,
4, 5, 6, 7, 8, or we shall write only is . For brief denota-
tions, we shall write 0u = (0x, 0y); 1u = (0u, 1y) =
(0x, 0y, 1y), 2u = (1u, 2y) = (0x, 0y, 1y, 2y) and
3u = (2u, 3y) = (0x, 0y, 1y, 2y, 3y). Here we note
in modern gravity, the so-called ADM (Arnowitt–Deser–
Misner) formalism is largely used with 3 + 1 splitting, or
any n + 1 splitting; see for details [52]. For the purposes of
this work, such splittings are not convenient because it is not
possible to elaborate a technique of general decoupling of
the gravitational field equations and generating off-diagonal
solutions. In these cases, the conventional one-dimensional
“fibers” result in certain degenerate systems of equations.
To construct exact solutions in four–ten-dimensional theo-
ries, it is more convenient to work with a correspondingly
defined non-integrable 2 + 2 + ··· splitting; see details in
[39,40,43]. For the heterotic supergravity, such geometric
constructions are explored in greater detail in [48]. In order
to connect “shell by shell” indices and coordinates to stan-
dard ones for supergravity theories (see [1,2]), we can con-
sider small Greek indices without sub indices, and respective
coordinates xμ, when α,μ, ... = 0, 1,... , 9. The identifi-
cation of shell coordinates with the standard ones follows
such a rule: x0 = u3 = t (time-like coordinate) and (for
space-like coordinates): x1 = u1, x2 = u2, x3 = u4, x4 =
u5, x5 = u6, x6 = u7, x7 = u8, x8 = u9, x9 = u10.
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Eur. Phys. J. C (2017) 77:17
On M, we can consider local frames/bases, eαs =
e
αs
αs (
su)∂/∂uαs , where the partial derivatives ∂βs := ∂/∂uβs
define local coordinate bases. We shall underline indices if
it is necessary to emphasize that such values are defined
with respect to a coordinate frame. The (co) frames, eαs =
e αsαs (
su)duαs , can be defined as dual to respective eαs .
For our purposes, it is convenient to work with nonholo-
nomic (non-integrable) distributions defining a 2 + 2 + ···
spacetime splitting. Such a distribution can be introduced to
define a nonlinear connection, N-connection, structure via a
Whitney sum
sN : TM= 0hM⊕0vM⊕ 1vM⊕2vM⊕3vM.
(2)
This formula states a conventional horizontal (h) and vertical
(v) “shell by shell” splitting. We shall use boldface letters in
order to define spaces and geometric objects enabled/adapted
to a N-connection structure. In local form, an N-connection
is defined by its coefficients Nas
is
when sN = Nas
is
(su)dxis ⊗
∂/∂yas .
A nonholonomic manifold is enabled with a nonholo-
nomic distribution of type (2), when (for instance, for the
“zero” shell) V hM ⊕ vM. The term N-anholonomic
manifold is also used. This definition comes form the fact
that the N-connection coefficients determine a system of N-
adapted local bases, with N-elongated partial derivatives,
eνs = (eis , eas ), and cobases with N-adapted differentials,
eμs = (eis , eas ), For s = 0,
ei0 = ei =
∂
∂xi0
− Na0
i0
∂
∂ya0
, ea0 =
∂
∂ya0
,
ei0 = dxi , ea = ea0 = dya + Nai dxi ; or/and
eis =
∂
∂xis
− Nas
is
∂
∂yas
, eas =
∂
∂yas
,
eis = dxis , eas = dyas + Nas
is
dxis
for s = 1, 2, 3.
(3)
The anholonomy relations
[eαs , eβs ] = eαs eβs − eβs eαs = W γs
αsβs
eγs
(4)
are computed Wbs
isas
= ∂as Nbs
is
and Was
js is
= as
is js
, where
the curvature of N-connection is defined as the Neijenhuis
tensor, as
is js
:= e js (Nas
is
)− eis (Nas
js
).
Any metric structure on M can be parametrized via non-
holonomic frame transforms as a distinguished metric (d-
metric, in boldface form), sg = {gαsβs },
sg = gis js (su) eis ⊗ e js + gasbs (su)eas ⊗ ebs
= gij (x) ei ⊗ e j + gab(u) ea ⊗ eb
+ ga1b1(1u) ea1 ⊗ eb1 + ··· + gasbs (su)eas ⊗ ebs .
(5)
Introducing dual coframes eμs = (eis , eas ) decomposed with
respect to N-elongated differentials duas as in (3), we obtain
a generic off-diagonal form
sg with respect to a dual coor-
dinate frame. A metric sg can be diagonalized by coordinate
transforms in a finite region if and only if the anholonomy
coefficients in (4) vanish, i.e. W γs
αsβs
= 0
A linear connection s D can be introduced on M in stan-
dard form. A distinguished connection, d-connection,
sD
is a linear connection preserving the N-connection spitting
under parallelism. One defines the curvatureRαsβs and torsion
T αs in standard forms; see details in [48]. In this paper, we
shall work with two very important linear connection struc-
tures determined by the same metric structure. These linear
connections are uniquely defined following the geometric
conditions:
sg →
{ s∇ :
s∇ (sg) = 0; s∇T = 0,
the Levi-Civita connection;
sD̂ :
sD̂ (sg) = 0; hT̂ = 0, 1vT̂ = 0, 2vT̂ = 0, 3vT̂ = 0. the canonical d-connection.
(6)
Here we note that the LC-connection s∇ = {  αsβsγs } can be
introduced without any N-connection structure. It can always
be canonically distorted to a necessary type of d-connection
sD completely defined by sg following certain geometric
principles. The canonical d-connection sD̂ is characterized
by a nonholonomically induced torsion d-tensor which is
completely defined by sg for any chosen sN = {Nas
is
}. The
N-adapted coefficients are parameterized “shell by shell” by
formulas
sT̂ = {T̂γsαsβs :T̂ isjs ks =L̂
is
js ks
−L̂isks js ,T̂
is
jsas
=Ĉis
jsbs
,
T̂ as
js is
= −asjs is ,T̂
cs
as js
=L̂cs
as js
− eas (Ncs
js
),
T̂ asbscs =Ĉ
as
bscs
−Ĉas
csbs
}.
(7)
It should be noted that this torsion is different form torsions in
Einstein–Cartan gauge type and string gravity theories with
absolutely antisymmetric torsion. Additional sources are not
necessary because a d-torsion (7) is determined by the non-
holonomic structures. In generalized theories, the torsion
fields which are independent from the metric and vielbein
fields may posses proper sources. Considering additional
assumptions, we can relate the values, (7) for instance, to
a subclass of nontrivial coefficients of an absolute antisym-
metric torsion in string gravity. We can always extract LC-
configurations with zero torsion if we additionally impose
for (7) the conditions
T̂γsαsβs = 0,
i.e.
sD̂|T̂ =0 → s∇.
(8)
Such additional nonholonomic constraints may be stated
in non-explicit forms and without certain limits with small
parameters and smooth functions. It should be noted that, in
123
Eur. Phys. J. C (2017) 77:17
Page 7 of 32
17
general, W γs
αsβs
(4) may not be zero even if the conditions (8)
are satisfied.
Any (pseudo-) Riemannian geometry can be equivalently
formulated in nonholonomic variables (sg, sN, sD̂) or using
the standard data (sg, s∇). Because both linear connections
s∇ and sD̂ are defined by the same metric structure, there
is a canonical distortion relation
sD̂ = s∇ + sẐ.
(9)
The N-adapted coefficients of a curvature d-tensorRαsβs =
{Rαsβsγsδs } of the canonical d-connection sD̂ and sg can be
computed, respectively, for all shells s = 0, 1, 2, 3; see for
details [48]. The Ricci d-tensorR̂ic = {R̂αsβs :=R̂τsαsβsτs }
is a respective contracting of the coefficients of curvature
tensor,
R̂αsβs =
{
R̂hs js :=R̂ishs js is ,
R̂ jsas := −P̂isjs isas ,
R̂bsks :=P̂asbsksas ,R̂ bscs =Ŝ
as
bscsas
}
.
(10)
Considering the inverse d-metric to sg, we define and com-
pute the scalar curvature of sD̂,
sR̂ := gαsβsR̂αsβs = gis jsR̂is js + gasbsR̂asbs
=R̂ +Ŝ + 1Ŝ + 2Ŝ + 3Ŝ,
(11)
with respective h- and v-components of the scalar curva-
ture,R̂ = gijR̂ij , S = gabSab, 1S = ga1b1Sa1b1 , 2S =
ga2b2 Sa2b2 ,
3S = ga3b3 Ra3b3 .
2.2 The AFDM for heterotic supergravity
We develop the “anholonomic frame deformation method”,
AFDM, and apply these geometric techniques for applica-
tions to 10-d gravity and heterotic supergravity formulated
in nonholonomic variables.
The heterotic supergravity field equations were formu-
lated in N-adapted form in [48]. They include terms of order
α′, equivalent to the equations of motion of heterotic non-
holonomic supergravity considered in [1]. In explicit form,
R̂μsνs + 2(sD̂d̂φ̂)μsνs −
1
4
ĤαsβsμsĤ
αsβs
νs
+ α
′
4
[
R̃μsαsβsγsR̃
αsβsγs
νs
− tr (̂FμsαsF̂ αs
νs
)] = 0,
(12)
sR̂ + 4̂φ̂ − 4|̂dφ̂|2 − 1
2
|Ĥ|2 + α
′
4
tr
[
|R̃|2 − |̂F|
]
= 0,
(13)
e2φ̂ d̂̂∗(e−2φ̂F̂)+Â ∧∗̂F̂−∗̂F̂ ∧Â+∗̂Ĥ ∧F̂ = 0,
(14)
d̂̂∗(e−2φ̂Ĥ) = 0,
(15)
where the Hodge operator∗̂, sD̂ = {D̂μs }, the canonical non-
holonomic d’Alembert wave operator̂ :=ĝμsνsD̂μsD̂νs ,
the Ricci d-tensorR̂μsνs , and scalar
sR̂ are determined by
a d-metricĝ (5). The curvature d-tensorR̃μsαsβsγs is taken
for an almost-Kähler structure on shells s = 1, 2, 3 as we
described above. The gauge field corresponds to the N-
adapted operator sAD̂ and curvatureF̂ = F(1ψ) via a map
constructed in [1,48]).
Equation (12) can be written as effective Einstein equa-
tions for the canonical d-connection sD̂,
sR̂ βsδs = ϒβsδs ,
(16)
L̂cs
as js
= eas (Ncs
js
),
Ĉis
jsbs
= 0, asjs is = 0.
(17)
The sources ϒβsδs can be formally defined as in GR but for
extra dimensions and in N-adapted form, when
ϒβsδs → 
(
Tβsδs
1
2
gβsδs
sT
)
for sD̂ → s∇.
The system (16) can be derived and formulated in variational
N-adapted form by considering a nonholonomic gravitational
Lagrange density of type gL̂ =R̂ and an effective Lagrange
density for matter mL̂. For simplicity, we shall consider
Lagrangians depending only on the coefficients of metric
field and matter field but not on their derivatives (for such
configurations, it will be possible to construct exact solu-
tions in explicit form). The energy-momentum d-tensor is
computed by the definition,
mT̂αβ :=−
2
√| sg|
δ(
√| sg| mL̂)
δ sgαβ
=mL̂ sgαβ + 2δ(
mL̂)
δ sgαβ
,
(18)
for | sg| = det | sgμν |. We conclude that by following an
N-adapted variational calculus with action
gS+ mS =
∫
d4u
√| sg|(gL̂ + mL̂),
we can elaborate a 10-d nonholonomic gravity theory with
gravitational field equations (16). A changing of geomet-
ric data (sg, sD̂) → (sg, s∇) is possible via the canonical
distorting relations (9), or imposing the zero torsion condi-
tion at the endT̂ = 0 (8) for extracting LC-configurations
sD̂|T̂ =0 = s∇ stated by (17). The matter Lagrange density
mL̂ can be chosen in such a form that, via corresponding
frame transforms, the sourceϒβsδs will encode the terms with
the effective matter fieldsφ̂,Ĥαsβsμs , interior space fields
R̃μsαsβsγs , and gauge fieldsF̂μsαs .
The nonholonomic equations of motion in heterotic string
gravity (12) can be written in the form (16) with an effec-
tive source (related via nonlinear transforms of generating
functions to certain effective cosmological constants),
ϒμsνs = φϒμsνs + Hϒμsνs + Fϒμsνs
+ Rϒμsνs , where
(19)
φϒμsνs = −2( sD̂d̂φ̂)μsνs with effective constant φ;
(20)
Hϒμsνs =
1
4
ĤαsβsμsĤ
αsβs
νs
with effective constant H; (21)
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17 Page 8 of 32
Eur. Phys. J. C (2017) 77:17
Fϒμsνs =
α′
4
tr
(̂
FμsαsF̂
αs
νs
)
with effective constant F; (22)
Rϒμsνs = −
α′
4
R̃μsαsβsγsR̃
αsβsγs
νs
with effective constantR̃.
(23)
The traces of the above sources, and the respective effective
cosmological constants, are related via the condition (13).
2.3 Decoupling and integration of nonholonomic equations
of motion
We show how the heterotic string gravitational field equa-
tions (12) written in the form (16) with sources (19) and
possible constraints (17), can be formally integrated in very
general forms for generic off-diagonal metrics with coeffi-
cients depending on all spacetime coordinates. Additional
conditions for extracting LC-configurations with s∇ will be
analyzed at the end after certain classes of general solutions
have been constructed.
2.3.1 Ansatz for metrics, N-connections, and gravitational
polarizations
In the simplest form, the decoupling property for any shell
s = 0, 1, 2, 3 can be proven for certain ansatz with at least
one Killing symmetry on the corresponding shell. Using an
additional conformal transform on the necessary shells, we
can extend the constructions for non-Killing configurations.
Such a procedure is described for arbitrary finite shell s in
Refs. [40–43]. In this work, we state the formulas for 10-d
gravity and heterotic string theory in explicit form when the
prime solutions are stationary ones, i.e., do not depend on
the time-like coordinate y3 = t.
Let us consider metrics of type (5), which via frame trans-
forms gαsβs = eα
′
s
αs e
β ′s
βs
gα′sβ ′s can be parametrized
5
s
K g = gi (xk)dxi ⊗ dxi + ha(xk , y4)ea ⊗ eb
+ ha1(uα, y6) ea1 ⊗ ea1 + ha2 (uα1 , y8) ea2
⊗ eb2 + ha3( uα2 , y10)ea3 ⊗ ea3 ,
(24)
where ea = dya + Nai dxi ,
for N3
i = ni (xk , y4), N4
i = wi (xk , y4);
ea1 = dya1 + Na1
α du
α,
for N5α = 1nα(uβ, y6), N6α = 1wα(uβ, y6);
ea2 = dya2 + Na2
α1 du
α1 ,
for N7α1 = 2nα1(uβ1 , y8), N8α1 = 2wα1 (uβ1 , y8);
ea3 = dya3 + Na3
α2 du
α2 ,
for N9α2 = 3nα2 (uβ2 , y10), N10
α2
= 3wα3 (uβ2 , y10).
5 We shall put a left label K in order to emphasize that this is a d-metric
with Killing symmetry.
Such an ansatz has a Killing vector ∂/∂y9 because the coor-
dinate y9 is not contained in the coefficients of such metrics.
We propose that via coordinate transforms we can eliminate
dependence on y3 = t and can parametrize h4 = 1 and
has =
h(y4)ȟas (x
i , ybs ) for shells s = 1, 2, 3, as it was
considered in [48], if the configurations are with warping on
y4.The ansatz (24) can be considered as a target d-metric of a
prime d-metric with flat domain wall considered in that asso-
ciated paper. In this work, we restrict our considerations to
stationary solutions in heterotic string gravity (which do not
depend on t). Inhomogeneous and locally anisotropic cos-
mological configurations in such string models, with generic
dependence on t (see examples in [26–30]) will be studied
in our future publications.
2.3.2 Ricci d-tensors and N-adapted sources
We suppose that via frame transforms it is always pos-
sible to introduce frame and coordinate parametrizations
for ansatz (24) with6 ∂4h3 = h∗3
= 0,∂6h5 = h∗1
5
=
0,∂8h7 = h∗2
7
= 0,∂10h9 = h∗3
9
= 0. In brief, the partial
derivatives are denoted, for instance, ∂1h = ∂h/∂x1 = h·,
∂2h = ∂h/∂x2 = h′,∂3h = 0, and ∂44h = ∂2h/∂y4∂y4 =
h∗∗,∂66h = ∂2h/∂y6∂y6 = h∗1∗1 , etc. We shall write
explicitly ∂5h = ∂h/∂y5, ∂6h = ∂h/∂x6,... without intro-
ducing “dot” and “prime” symbols for partial derivatives on
shells s = 1, 2, 3 but working with “star” partial derivatives
on these shells, considering, respectively, ∗1, ∗2, ∗3 if neces-
sary written as ∗s . A tedious computation of the coefficients
of the canonical d-connectionD̂ = {̂γsαsβs } for the ansatz
(24) and then of the corresponding nontrivial coefficients of
the Ricci d-tensorR̂αsβs (10), see similar details in [35,38–
43], results in nontrivial N-adapted coefficients:
R̂11 =R̂22 = −
1
2g1g2
[
g··2 −
g·1g·2
2g1
− (g
·
2)
2
2g2
+ g′′1 −
g′1g′2
2g2
−
(
g′1
)2
2g1
]
,
(25)
R̂33 =R̂44 = −
1
2h3h4
[
h∗∗
3 −
(
h∗3
)2
2h3
− h
∗
3h
∗
4
2h4
]
,
(26)
R̂3k = h3
2h4
n∗∗
k +
(
h3
h4
h∗4 −
3
2
h∗3
)
n∗k
2h4
,
(27)
R̂4k = wk
2h3
[
h∗3 −
(
h∗3
)2
2h3
− (h
∗
3)(h
∗
4)
2h4
]
+ h
∗
3
4h3
(
∂kh3
h3
+ ∂kh4
h4
)
− ∂k(h
∗
3)
2h3
;
(28)
6 We can construct special classes of solutions if such conditions are
not satisfied.
123
Eur. Phys. J. C (2017) 77:17
Page 9 of 32
17
on shell s = 1 with τ = 1, 2, 3, 4,
R̂55 =R̂66 = −
1
2h5h6
[
h∗1∗1
5 −
(h∗1
5 )
2
2h5
− h
∗1
5 h
∗1
6 )
2h6
]
, (29)
R̂5τ = h5
2h6
1n∗1
τ +
(
h5
h6
h∗1
6 −
3
2
h∗1
5
) 1n∗1
τ
2h6
,
(30)
R̂6τ =
1wτ
2h5
[
h∗1∗1
5 −
(
h∗1
5
)2
2h5
− h
∗1
5 h
∗1
6
2h6
]
+ h
∗1
5
4h5
(
∂τh5
h5
+ ∂τh6
h6
)
− ∂τ (h
∗1
5 )
2h5
,
(31)
on shell s = 2 with τ1 = 1, 2, 3, 4, 5, 6:
R̂77 =R̂88 = −
1
2h7h8
[
h∗2∗2
7 −
(
h∗2
7
)2
2h7
− h
∗2
7 h
∗2
8
2h8
]
,
R̂7τ1 =
h7
2h8
2n∗2∗2
τ1
+
(
h7
h8
h∗2
8 −
3
2
h∗2
7
) 2n∗2
τ1
2h7
,
R̂8τ1 =
2wτ1
2h7
[
h∗2∗2
7 −
(
h∗2
7
)2
2h7
− h
∗2
7 h
∗2
8
2h8
]
+ h
∗2
7
4h7
(
∂τ1h7
h7
+ ∂τ1h8
h8
)
− ∂τ1(h
∗2
7 )
2h7
,
(32)
on shell s = 3 with τ2 = 1, 2, 3, 4, 5, 6, 7, 8:
R̂99 =R̂10
10 = −
1
2h9h10
[
h∗3∗3
9 −
(
h∗3
9
)2
2h9
− h
∗3
9 h
∗3
10)
2h10
]
,
R̂9τ2 =
h9
2h10
2n∗3∗3
τ2
+
(
h9
h10
h∗3
10 −
3
2
h∗3
9
) 2n∗3
τ2
2h9
,
R̂10τ2 =
2wτ1
2h9
[
h∗3∗3
9 −
(
h∗3
9
)2
2h9
− h
∗3
9 h
∗3
10
2h10
]
+ h
∗3
9
4h9
(
∂τ2h9
h9
+ ∂τ2h10
h10
)
− ∂τ2(h
∗3
9 )
2h9
.
(33)
Using the above formulas, we can compute the Ricci scalar
(11) for sD̂ for the ansatz (24) and s = 0, 1, 2, 3:
0R̂ = 2(R̂11 +R̂33), 1R̂ = 2(R̂11 +R̂33 +R̂55),
2R̂ = 2(R̂11 +R̂33 +R̂55 +R̂77),
3R̂ = 2(R̂11 +R̂33 +R̂55 +R̂77 +R̂99).
This imposes certain N-adapted symmetries on the Einstein
d-tensor for the ansatz (24); see for details [40,43].
2.3.3 N-adapted sources and nonholonomically modified
Einstein equations
We will be able to integrate nonholonomic equations of
motion in heterotic string gravity in explicit form for very
general assumptions if the source ϒβsδs (19) is parametrized
in N-adapted form. This can be determined by five indepen-
dent effective sources, respectively, on h- and sv-subspaces,
which will be related to certain effective cosmological con-
stants corresponding to the formulas
ϒ̂11 =ϒ̂22 = hϒ(xk) → h = φh+ Hh + Fh + Rh ,
ϒ̂33 =ϒ̂44 =ϒ(xk, y4) → =φ+ H+ F+ R,
ϒ̂55 =ϒ̂66 = 1ϒ(xk, ya, y6) → 1
= φ1+ H1 + F1 + R1 ,
ϒ̂77 =ϒ̂88 = 2ϒ(xk, ya, ya1 , y8) → 2
= φ2+ H2 + F2 + R2 ,
ϒ̂99 =ϒ̂10
10 = 3ϒ(xk, ya, ya1 , ya2 , y10) → 3
= φ3+ H3 + F3 + R3 .
(34)
For certain general configurations, it will be possible to fix
generating and effective sources of type h =
s = 
for all s, when a value for the corresponding contribution
of fields can be zero, or nonzero, for instance, φh = 0
but H2 
= 0. This depends on the type of vacuum, non-
vacuum, or effective vacuum model we study. It is possible
to compensate for contributions into an effective source of a
field with contributions of another field, for instance, to get
 = φ + H + F + R = 0 even when not all
the values of such effective cosmological constants are zero.
We note that by prescribing certain values of sources (34) we
can relate via nonholonomic frame transforms (in coordinate
and/or N-adapted form)ϒαsβs = eα
′
s
αs e
β ′s
βs
ϒ̂α′sβ ′s ,whereϒαsβs
is any (effective) source (19), in heterotic string gravity, or
(18), in a 10-d nonholonomic generalization of GR. For any
effective mL̂ we can solve a system of quadratic algebraic
equations for e
α′s
αs in a form compatible with transforms to
N-adapted frames of the metric/d-metric components,
In this section, we shall construct general solutions of the
generalized nonholonomic Einstein equations (16) with Ricci
d-tensors (25)–(33) and effective sources (encoding contri-
butions from heterotic supergravity) (19) parameterized in
the form (34) written in N-adapted form as
R̂11 =R̂22 = − hϒ(xk),R̂33 =R̂44 = − ϒ(xk, y4),
R̂55 =R̂66 = − 1ϒ(xk, ya, y6),
R̂77 =R̂88 = − 2ϒ(xk, ya, ya1 , y8),
R̂99 =R̂10
10 = − 3ϒ(xk, ya, ya1 , ya2 , y10).
(35)
Similar equations can be written recurrently for arbitrary
finite extra dimensions.
2.3.4 The ansatz for effective matter fields in heterotic
string gravity
Assumptions on nontrivialφ̂-configurations We shall find d-
metrics of type (5) for which the contributions of the field
φ̂ can be included by nonholonomic deformationsg̊ → sg
into certain generic off-diagonal terms, i.e. into N-connection
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17 Page 10 of 32
Eur. Phys. J. C (2017) 77:17
coefficients. For instance if under a prime configuration
0φ̂(xi , ya, yas ) → ηφ̂ =φ̂(xi , y4) for whichd̂φ̂ = 0.
This equation is equivalent to a linear system of equations,
∂iφ̂ − wi (xk, y4)φ̂∗ = 0,
(36)
which can be solved in explicit form if the N-connection
coefficients wi are defined (see next sections for how such
values can be found in explicit form). For such configura-
tions, we state that φϒμsνs and
φ in (19) and (20) are zero
and the fieldsφ̂ with such configurations contribute to possi-
ble heterotic supergravity effects only via possible nontrivial
off-diagonal interactions and not via effective sources. The
terms withφ̂ also vanish in all nonholonomic equations of
motion (12)–(15) written with respect to N-adapted frames if
the conditions (36) are satisfied. Nontrivial coefficients with
φ̂ and its partial derivatives will appear if certain physical
equations are written, for instance, in coordinate frames. We
have chosen a special type of configurations forφ̂ in order to
simplify the procedure of finding generic off-diagonal solu-
tions in heterotic string gravity in explicit form.
Nonholonomic gauge configurations In general form, we
can consider any Hϒμsνs ,
Fϒμsνs and
Rϒμsνs for which
ϒμsνs = Hϒμsνs + Fϒμsνs + Rϒμsνs , can be nonholonom-
ically transformed into an N-adapted diagonal form
ϒμsνs = diag[ hϒ(xk), hϒ(xk), ϒ(xk , y4),
ϒ(xk , y4), 1ϒ(x
k, ya, y6), 1ϒ(x
k , ya, y6),
2ϒ(x
k , ya, ya1 , y8), 2ϒ(x
k , ya, ya1 , y8),
3ϒ(x
k , ya, ya1 , ya2 , y10), 3ϒ(x
k , ya, ya1 , ya2 , y10)].
Nonholonomic deformations of string gauge fields can be
parametrized in the form
Ĥαsβsμs = 0Ĥαsβsμs + ηĤαsβsμs ,F̂μsαs = 0F̂μsαs
+ ηF̂μsαs ,
R̃μsαsβsγs = 0R̃μsαsβsγs + ηR̃μsαsβsγs .
where, respectively,
Hϒμsνs = diag[ Hh ϒ(xk), Hh ϒ(xk), Hϒ(xk, y4),
Hϒ(xk, y4), H1 ϒ(x
k, ya, y6), H1 ϒ(x
k, ya, y6),
H
2 ϒ(x
k, ya, ya1 , y8), H2 ϒ(x
k, ya, ya1 , y8),
H
3 ϒ(x
k, ya, ya1 , ya2 , y10),
H
3 ϒ(x
k, ya, ya1 , ya2 , y10)];
Fϒμsνs = diag[ Fh ϒ(xk), Fh ϒ(xk), Fϒ(xk, y4),
Fϒ(xk, y4), F1 ϒ(x
k, ya, y6), F1 ϒ(x
k, ya, y6),
F
2 ϒ(x
k, ya, ya1 , y8), F2 ϒ(x
k, ya, ya1 , y8),
F
3 ϒ(x
k, ya, ya1 , ya2 , y10),
F
3 ϒ(x
k, ya, ya1 , ya2 , y10)];
Rϒμsνs = diag[ Rh ϒ(xk), Rh ϒ(xk), Rϒ(xk, y4),
Rϒ(xk, y4), R1 ϒ(x
k, ya, y6), R1 ϒ(x
k, ya, y6),
R
2 ϒ(x
k, ya, ya1 , y8), R2 ϒ(x
k, ya, ya1 , y8),
R
3 ϒ(x
k, ya, ya1 , ya2 , y10),
R
3 ϒ(x
k, ya, ya1 , ya2 , y10)].
(37)
For instance, kink-like almost-Kähler configurations can be
encoded as a prime configuration 0Ĥαsβsμs modifying for the
target configurationĤαsβsμs the solutions for off-diagonal
metrics via an effective source.
Such parametrizations are possible by considering the
ansatz with effective cosmological constants
Ĥαsβsμs = Hs
√|gβsμs |αsβsμs ,
F̂μsαs = Fs
√|gβsμs |μsαs ,
R̃μsαsβsγs = Rs
√|gβsμs |μsαsβsγs ,
with absolute antisymmetric -tensors (we refer the reader
to similar details for the 4-d case to [42] and the references
therein). For such an ansatz, we obtain effective energy-
momentum tensors
Hϒμsνs =−
10
2
( Hs)2gβsμs ,
for H = −5( Hs)2,
Fϒμsνs =−
10α′
2
nF (
Fs)2gβsμs ,
for F=−5nF ( Fs)2,
Rϒμsνs =
10α′
2
nR(
Rs)2gβsμs ,
for R=−5trnR( Rs)2,
(38)
were we considered Eqs. (34) and (19)–(23) and, for instance
the numbers nF = tr[internal F] and nR = tr[internalR̃]
depends on the representation of the Lie algebra for F and
on the representation of Lie groups on the internal space.
2.3.5 Decoupling of nonholonomic equations of motion
and effective
Considering the ansatz (24) for gi (xk) = i eq(xk ),i = 1, in
(35) with respective sources, we obtain this nonlinear system
of PDEs:
q ·· + q ′′ = 2 hϒ,
(39)
 ∗h∗3 = 2h3h4 ϒ,
(40)
n∗∗
i + γ n∗
i = 0,
(41)
βwi − αi = 0,
(42)
1 ∗1h∗1
5 = 2h5h6 1ϒ,
(43)
1n∗1∗1
i1
+ 1γ 1n∗1
i1
= 0,
(44)
1β 1wi1 − 1α1 = 0,
(45)
2 ∗2h∗2
7 = 2h7h8 2ϒ,
2n∗2∗2
i2
+ 2γ 2n∗2
i2
= 0,
2β 2wi2 − 2αi2 = 0,
(46)
3 ∗3h∗3
7 = 2h9h10 3ϒ,
3n∗3∗3
i3
+ 3γ 3n∗3
i3
= 0,
3β 3wi3 − 3αi3 = 0.
(47)
123
Eur. Phys. J. C (2017) 77:17
Page 11 of 32
17
In equivalent form, the same equations are obtained recur-
rently if we write, for instance, swis instead of
swτs−1,
snis
instead of snτs−1 etc. (some equations and solutions can be
parametrized in more simple forms if we follow the first con-
vention, other equations will be more “compact” if we follow
the second convention). In these equations, the generating
functions
 = ln ||,
1 = ln |
1|,
2 = ln | 2|,
3 = ln | 3|,
(we shall use this in the next sections, on convenience formu-
las with  - and/or -values) and the α-, β-, γ -coefficients
on corresponding shells are defined, respectively:
 = ln
∣∣∣∣
h∗3
√|h3h4|
∣∣∣∣ , γ :=
(
ln
|h3|3/2
|h4|
)∗
,
αi = h
∗
3
2h3
∂i, β = h
∗
3
2h3
 ∗,
(48)
1 = ln
∣∣∣∣∣
h∗1
5
√|h5h6|
∣∣∣∣∣ ,
1γ :=
(
ln
|h5|3/2
|h6|
)∗1
,
1ατ = h
∗1
5
2h5
∂τ
1,
1β = h
∗1
5
2h5
∂τ
1,
2 = ln
∣∣∣∣ h
∗2
7
√|h7h8|
∣∣∣∣ ,
2γ :=
(
ln
|h7|3/2
|h8|
)∗2
,
2ατ1 =
h∗2
7
2h7
∂τ1
2,
2β = h
∗2
7
2h7
∂τ1
2,
3 = ln
∣∣∣∣ h
∗3
9
√|h7h8|
∣∣∣∣ ,
3γ :=
(
ln
|h9|3/2
|h10|
)∗3
,
3ατ2 =
h∗3
9
2h9
∂τ2
3,
3β = h
∗3
9
2h9
∂τ2
3,
when the frame/coordinate systems are chosen in such a way
that nonzero conditions for the partial derivatives are satis-
fied:  ∗, h∗a, 1 ∗1 , h
∗1
a1 ,
2 ∗2 , h∗2
a2 ,
3 ∗3 , h∗3
a3
= 0 (this
allows us to avoid singular nonholonomic deformations).
Equations (39)–(47) prove a very important decoupling
property of the heterotic string equations (12) and 4d–10d
(generalized) Einstein equations (16) with respect to corre-
spondingly N-adapted frames. In explicit and in certain sim-
ple forms, such formulas can be obtained for metrics with at
least one Killing symmetry. Nevertheless, the constructions
can be generalized for non-Killing configurations in any finite
extra-dimension gravity; see a corresponding technique in
[40,43]. In brief the decoupling property of the AFDM can
be explained for the 4-d configurations:
1. Equation (39) is just a 2-d Laplacian, which can be solved
for any h-source hϒ(xk).
2. Equation (40) contains only the partial derivative ∂4,
equivalently ∗-derivative, and constraints by a system of
two equations, together with the formula for the value 
in (48), four values h3(xi , y4), h4(xi , y4) and (xi , y4)
and source ϒ(xk, y4). Prescribing any two such func-
tions, we can define (integrating on y4) two other such
functions. We note that ha are coefficients of a d-metric,
 is a generating function, and ϒ is related to extra-
dimensional string contributions.
3. Using h3 and , or , from the previous point, we com-
pute the coefficients αi and β; see (48). This allows us to
define wi from the algebraic equations (42).
4. Having computed the coefficientγ (48), the N-connection
coefficients ni can be defined after two integrations on
y4 in (41).
We can repeat the steps 2–4 recurrently on shells s =
1, 2, 3, adding dependencies on 2 + 2 + 2 extra-dimension
coordinates, respectively. In this way, we involve new classes
of generating functions and effective sources. Solving the
systems (43)–(47) recurrently, we can compute all d-metric
and N-connection coefficients for a 10-d ansatz (5).
2.3.6 Integration of nonholonomic equations of motion by
generating functions and effective sources
The system of nonlinear PDEs (39)–(47) with coefficients
(48) and effective sources (37) with contributions from string
gravity can be integrated in general forms on any shell up to
10-d.
4-d non-vacuum configurations The coefficients gi = eq(xk )
are defined by solutions of the corresponding 2-d Poisson
equation (39) as we mentioned above (see point 1 at the end
of previous subsection).
We can integrate the system of nonlinear PDEs consist-
ing of the first equation in (48) and (40) for arbitrary source
ϒ(xk, y4) and with generating function (xk, y4) = e .
The solutions will be generated for a stationary d-metric
when the coefficients do not depend on time-like coordinate
y3 = t, when ∗ = 0. The vertical effective gravitational
field equations (40)–(42) transform, respectively, into
∗ h∗3 = 2 h3h4 ϒ,
(49)
√|h3h4| = h∗3,
(50)
n∗∗
i +
(
ln | h3|3/2/|h4|
)∗
n∗
i = 0,
(51)
∗wi − ∂i = 0.
(52)
This system can be solved in very general forms by prescrib-
ing ϒ and  and integrating the equations “step by step”.
Introducing the function
ρ2 := −h3h4
(53)
(the sign − is motivated by the pseudo-Euclidean signature),
we express (49) and (50) as
∗h∗3 = −2ρ2ϒ and h∗3 = ρ.
(54)
123
17 Page 12 of 32
Eur. Phys. J. C (2017) 77:17
Using h∗3 from the second equation (54) in the first equation,
we write
ρ = −1
2
∗
ϒ
.
(55)
This value, together with the second equation of (54) and a
further integration on y4, result in
h3 = h[0]
3 (x
k)− 1
4
∫
dy4
(2)∗
ϒ
,
(56)
where h[0]
3 (x
k) is an integration function. Considering Eqs.
(55), (53), and (56), we compute
h4 = − 1
4h3
(
∗
ϒ
)2
= −1
4
(
∗
ϒ
)2 (
h[0]
3 −
1
4
∫
dy4
(2)∗
ϒ
)−1
.
(57)
The first parts of the N-connection coefficients are found by
integrating two times on y4 in (51) written in the form
n∗∗
i = (n∗
i )
∗ = − n∗
i (ln |h3|3/2/|h4|)∗
for the coefficient γ defined in (48). Integrating two times on
y4 for explicit values of (57) and (56), we compute
nk(x
i , y4) = 1nk + 2nk
∫
dy4
h4
| h3|3/2
= 1nk + 2ñk
∫
dy4
( 2)∗
| h3|5/2ϒ2
= 1nk + 2nk
∫
dy4
(2)∗
ϒ2
×
∣∣∣∣h[0]
3 −
1
4
∫
dy4
(∗)2
ϒ
∣∣∣∣
−5/2
,
containing also a second set of integration functions 1nk(xi )
and the redefined second integration function 2ñk(xi ).
We can solve the linear algebraic equations (52) and
express
wi = ∂i /∗.
Putting together all the above formulas and writing the
effective source (34) in explicit form, we obtain the formulas
for the coefficients of a d-metric and a N-connection deter-
mining a class of stationary solutions for the system (39)–
(42),
gi = gi [ q, hϒ, ] = e q(xk)
as a solution of 2-d Poisson equations (39);
h3 = h[0]
3 (x
k)− 1
4
∫
dy4
(2)∗
ϒ
;
h4 = −1
4
(
∗
ϒ
)2 (
h[0]
3 −
1
4
∫
dy4
(2)∗
ϒ
)−1
;
nk = 1nk+ 2nk
∫
dy4
(2)∗
ϒ2
∣∣∣∣h[0]
3 −
1
4
∫
dy4
(2)∗
ϒ
∣∣∣∣
−5/2
;
wi = ∂i /∗.
(58)
Using coefficients (58), we define such a class of quadratic
elements for off-diagonal 4-d stationary configurations in
heterotic supergravity with nonholonomically induced tor-
sion,
ds2K4d = gαβ(xk, y4)duαduβ = eq [(dx1)2 + (dx2)2]
+
[
h[0]
3 (x
k)− 1
4
∫
dy4
(2)∗
ϒ
]
×
[
dt +
(
1nk + 2nk
∫
dy4
(2)∗
ϒ2
×
∣∣∣∣h[0]
3 −
1
4
∫
dy4
(2)∗
ϒ
∣∣∣∣
−5/2)
dxk
]2
− 1
4
(
∗
ϒ
)2 (
h[0]
3 −
1
4
∫
dy4
(2)∗
ϒ
)−1
×
[
dy4 + ∂i 
∗
dxi
]2
.
(59)
Such a class of metrics also contains exact solutions for
the canonical d-connectionD̂ in R2 gravity with effective
scalar field encoded into a nonholonomically polarized vac-
uum with a special parametrization of the source ϒ; see for
details [14,70].
Examples of 4-d–10-vacuum configurations The configura-
tions with zero source cannot be constructed as particular
cases of former off-diagonal solutions with ϒ = 0 because
these limits are not smooth for metrics (59). In string heterotic
gravity, such conditions can be satisfied if different effective
fields compensate their mutual contributions and result in an
effective vacuum gravitational configuration.
For the ansatz (24), we can construct solutions when the
nontrivial coefficients of the Ricci d-tensor (25)–(32) are zero
but the Ricci and torsion d-tensor are not trivial. The first
equation is a typical example of 2-d Laplace equation with
solutions expressed in the form gi = eq(xk ,ϒ=0).
In 4-d, there are three general classes of off-diagonal met-
rics which result in zero coefficients (26)–(28). Such con-
structions can be generalized for 10-d generic off-diagonals
with mixing standard vacuum configurations for the 4-d
spacetimes.
1. If h∗3 = 0 but h3
= 0, h∗4
= 0 and h4
= 0, we obtain
only one nontrivial equation (27),
n∗∗
k + n∗k (ln |h4|)∗ = 0,
123
Eur. Phys. J. C (2017) 77:17
Page 13 of 32
17
where h4(xi , y4) and wk(xi , y4) are arbitrary generating
functions. Integrating two times on y4, we obtain
nk = 1nk + 2nk
∫
dy4/h4
(60)
with integration functions 1nk(xi ) and 2nk(xi ). In 4-d,
this defines a quadratic line element
ds2v,4−d
= eq(xk ,ϒ=0)(dxi )2 + 0h3(xk)
×
[
dt + ( 1nk(xi )+ 2nk(xi )
∫
dy4/h4)dx
k
]2
+ h4(xi , y4)[dy4 + wi (xk, y4)dxi ]2.
Recurrently, we can construct effective stationary vac-
uum configurations in 10-d [with i1 = (1, 2, 3, 4); i2 =
(1, 2, 3, 4, 5, 6); i3 = (1, 2, 3, 4, 5, 6, 7, 8); a1 = 5, 6;
a2 = 7, 8; a3 = 9, 10],
ds2v,10−d = eq(x
k ,ϒ=0)(dxi )2
+
{
0h3(x
k)
[
dt+( 1nk(xi )+ 2nk(xi )
∫
dy4/h4)dx
k
]2
+ h4(xi , y4)[dy4 + wi (xk , y4)dxi ]2
}
type1,s=0
+ 0h5(xk , y4)
[
dy5 + ( 11nk1(xi , y4)
+ 12nk1(xi , y6)
∫
dy6/h6)dx
k1
]2}
type1,s=1
+
{
h6(x
i , y4, y6)[dy6 + 1wi1(xk , y4, y6)dxi1 ]2
+ 0h7(xk , y4, ya1)
[
dy7 + ( 21nk2 (xi , y4, ya1)
+ 22nk2 (xi , y4, ya1)
∫
dy8/h8)dx
k2
]2}
type1,s=2
+
{
h8(x
i , y4, ya1 , y8)[dy8 + 2wi2 (xk , y4, ya1 , y8)dxi2 ]2
+ 0h9(xk , y4, ya1 , ya2 )
[
dy9 + ( 31nk3(xi , y4, ya1 , ya2 )
+ 32nk3(xi , y4, ya1 , ya2 )
∫
dy10/h10)dx
k3
]2
+ h10(xi , y4, ya1 , ya2 , y10)
[
dy10
+ 3wi3(xk , y4, ya1 , ya2 , y10)dxi3
]2}
type1,s=3
.
For instance, the bracket {...}type1,s=2 states that the vac-
uum metric is of type 1 on shell 2. The coefficients of
such d-metrics do not depend on variables (t, y9) and, on
respective shells, the values
0h3, 1nk, 2nk,
0h5,
1
1nk1 ,
1
2nk1 ,
0h7,
2
1nk2 ,
2
2nk2 ,
0h9,
3
1nk3 ,
3
2nk3
are integration functions and q, h4,wi ,h6, 1wi1 , h8,
2wi2 , h10,
3wi3 are generating functions for 10-d non-
holonomic effective vacuum heterotic string configura-
tions. In the above formulas, h∗1
5 = 0, h5
= 0, h∗1
6
= 0,
and h6
= 0; h∗2
7 = 0, h7
= 0, h∗2
8
= 0, and
h10
= 0; h∗3
9 = 0, h9
= 0, h∗3
10
= 0, and h10
= 0.
2. In such cases, h∗3
= 0 and h∗4
= 0. It is possible to solve
the Eqs. (26) and (40) in a self-consistent form for ϒ =
0 if  ∗ = 0 for respective coefficients in (48). Fixing
 = 0 = const, we can consider arbitrary functions
wi (xk, y4) because β = αi = 0 for such configurations.
The conditions (48) are satisfied by any
h4 = 0h4(xk)[(
√|h3|)∗]2,
(61)
where 0h3(xk) is an integration function and h3(xk, y4)
is any generating function. The coefficients nk should be
found from (27); see (60). Such a family of 4-d vacuum
generic off-diagonal metrics is described by
ds2v,4d = eq(x
k ,ϒ=0)(dxi )2 + h3(xi , y4)
{
dt + (1nk(xi )
+ 2ñk(xi )
∫
dy4[(|h3|3/4)∗]2dxi
}2
+ 0h4(xk)[(
√|h3|)∗]2[dy4+wi (xk, y4)dxi ]2,
(62)
where the integration function 2ñk(xi ) contains certain
integration coefficients. We can extend such metrics on
any shell s = 1, 2, 3 preserving the conditions of a zero
effective source and adding respective generating func-
tions
s = 1 : 1 ∗1 = 0, h6 = 0h6(xk , y4)[(
√|h5|)∗1 ]2,
gener. functs.
{
h5(xi , y4, y6)
1wi1 (x
i , y4, y6)
;
s = 2 : 2 ∗2 = 0, h8 = 0h8(xk , y4, ya1 )[(
√|h7|)∗2 ]2,
gener. functs.
{
h7(xi , y4, ya1 , y8)
2wi2 (x
i , y4, ya1 , y8)
;
s = 3 : 3 ∗3 =0, h10= 0h10(xk , y4, ya1 , ya2 )[(
√|h9|)∗3 ]2,
gener. functs.
{
h9(xi , y4, ya1 , ya2 , y10)
3wi3 (x
i , y4, ya1 , ya2 , y10)
.
123
17 Page 14 of 32
Eur. Phys. J. C (2017) 77:17
In 10-d, the vacuum solutions (62) are generalized to
ds2v,10−d = eq (dxi )2
+ h3
{
dt + ( 1nk + 2ñk
∫
dy4[(|h3|3/4)∗]2dxk
}2
+ 0h4[(
√|h3|)∗]2[dy4 + widxi ]2
+ h5
{
dy5 + ( 11nk1 + 12ñk1
∫
dy6[(|h5|3/4)∗1 ]2dxk1
}2
+ 0h6[(
√|h5|)∗1 ]2[dy6 + 1wi1 dxi1 ]2
+ h7
{
dy7 + ( 21nk2 + 2ñk2
∫
dy8[(|h7|3/4)∗2 ]2dxk2
}2
+ 0h8[(
√|h7|)∗2 ]2[dy8 + 2wi2 dxi2 ]2
+ h9
{
dy9 + ( 31nk3 + 2ñk3
∫
dy10[(|h9|3/4)∗3 ]2dxk3
}2
+ 0h10[(
√|h9|)∗3 ]2[dy10 + 2wi3 dxi3 ]2,
for integration functions
0h4(x
k),1 nk(x
i ), 2ñk(x
i ); 0h6(xk , y4),
1
1nk1(x
i , y4), 12ñk1(x
i , y4);
0h8(x
k , y4, ya1), 21nk2 (x
i , y4, ya1),
2
2ñk2 (x
i , y4, ya1);
0h10(x
k , y4, ya1 , ya2 ), 31nk3(x
i , y4, ya1 , ya2 ),
3
2ñk2 (x
i , y4, ya1 , ya2 ),
In the above shell formulas, h∗1
5
= 0 and h∗1
6
= 0; h∗2
7
=
0 and h∗2
8
= 0; h∗3
9
= 0 and h∗3
10
= 0.
3. We consider that h∗3
= 0 but h∗4 = 0. Equation (26)
transforms into
h∗∗
3 −
(
h∗3
)2
2h3
= 0,
when the general solution is h3(xk, y4) =
[
c1(xk)+ c2
(xk)y4
]2
, with generating functions c1(xk), c2(xk), and
h4 = 0h4(xk). For  = 0 = const, we can take any
values wi (xk, y4) because β = αi = 0. The coefficients
ni are found from (27) and/or, equivalently, to (41) with
γ = 32 |h3|∗. Integrating on y4, this subclass of N-coefficients
are computed
ni = 1ni (xk)+ 2ni (xk)
∫
dy4|h3|−3/2 = 1ni (xk)
+ 2ñi (xk)[c1(xk)+c2(xk)y4]−2,
with integration functions 1ni (xk) and 2ni (xk), or redefined
2ñi = − 2ni/2c2. The quadratic line element for this class
of solutions for vacuum metrics is described by
ds2v,4−d = eq(dxi )2 + [c1(xk)+ c2(xk)y4]2[dt + ( 1ni (xk)
+ 2ñi (xk)[c1(xk)+ c2(xk)y4]−2)dxi ]2
+ 0h4(xk)[dy4 + wi (xk, y4)dxi ]2.
(63)
On extra shells, this type of nonholonomic vacuum solutions
are given by quadratic elements
ds2v,10−d3 = eq(dxi )2 + {[c1 + c2y4]2[dt
+ ( 1ni + 2ñi [c1 + c2y4]−2)dxi ]2
+ 0h4[dy4 + widxi ]2}type3,s=0 + {[ 1c1 + 1c2y6]2
×[dy5 + ( 11ni1 + 12ñi1 [ 1c1 + 1c2y6]−2)dxi1 ]2
+ 0h6[dy6 + 1wi1dxi1 ]2}type3,s=1
+{[ 2c1 + 2c2y8]2[dy7 + ( 21ni2 + 22ñi2 [ 2c1
+ 2c2y8]−2)dxi2 ]2 + 0h8[dy8 + 2wi2 dxi2 ]2}type3,s=2
+{[ 3c1 + 3c2y10]2[dy9 + ( 31ni3 + 32ñi3 [ 2c1
+ 2c2y10]−2)dxi3 ]2 + 0h10[dy10 + 3wi3dxi3 ]2}type3,s=3.
The generating functions are
q(xk), wi (x
k, y4),
1wi1(x
k, y4, y6),
2wi2(x
k, y4, ya1 , y8),
3wi3(x
k, y4, ya1 , ya2 , y10)
and the integration functions are
c1(x
k), c2(x
k), 1ni (x
k), 2ñi (x
k), 0h4(x
k);
1c1(x
k, y4), 1c2(x
k, y4), 11ni1(x
k, y4),
1
2ñi1(x
k, y4), 0h6(x
k, y4);
2c1(x
k, y4, ya1), 2c2(x
k, y4, ya1), 21ni2(x
k, y4, ya1),
2
2ñi2(x
k, y4, ya1), 0h8(x
k, y4, ya1);
3c1(x
k, y4, ya1 , ya2), 3c2(x
k, y4, ya1 , ya2),
3
1ni2(x
k, y4, ya1 , ya2), 32ñi2(x
k, y4, ya1 , ya2),
0h10(x
k, y4, ya1 , ya2).
Finally, we note that we can construct generic off-diagonal
vacuum solutions in heterotic supergravity with different
types on different shells
ds2vaccum = eq + {...}type,s=0 + {...}type,s=1
+{...}type,s=2 + {...}type,s=3
In certain cases, the type may be the same on two or three
shells, but the parametrizations of mixed types of nonholo-
nomic vacuum solutions is given by different dependencies
on shell coordinates of the generating and integration func-
tions.
Extra-dimensional non-vacuum solutions The solutions for
extra dimensions can be constructed in certain forms which
are similar to the 4-d ones with (58) using new classes of
generating and integration functions with dependencies on
extra dimensional coordinates. For instance, we can generate
solutions of the system (43)–(45) with coefficients (48)onthe
shell s = 1 following a formal analogy when ∂4 → ∂6, i.e.
123
Eur. Phys. J. C (2017) 77:17
Page 15 of 32
17
∗ → ∗1,(xk, y4) → 1(xk1 , y6), with 1(xk1 , y4) =
e
1 ; ϒ(xk, y4) → 1ϒ(xk1 , y6) ··· and applying the same
procedure as for the 4-d case but extending the number of
shell coordinates, respectively, for generating and integration
functions.
The solutions on s = 1 are also generated as station-
ary d-metrics when the coefficients do not depend on time-
like coordinate y3 = t and do not depend on the coordinate
y5. For nontrivial 1ϒ, we have to choose 1∗1
= 0. The
effective gravitational field equations (43)–(45) and respec-
tive equation for 1 from (48) transform (respectively) into
1∗1 h∗1
5 = 2 h5h6 1ϒ 1,
√|h5h6| 1 = h∗1
5 ,
1n∗1∗1
i1
+ (ln | h5|3/2/|h6|)∗1 1n∗1
i1
= 0,
1 1w
∗1
i1
− ∂i1 1 = 0.
Prescribing 1ϒ and 1, such equations for 6-d gravity can
be integrated as in the case of s = 0 with two vertical coordi-
nates in 4-d gravity, i.e. integrating equations “step by step”.
Introducing the function ( 1ρ)2 := h5h6, (for extra shells
the sign is different because we work with space-like coor-
dinates) we express the first two equations above as
1∗1h∗1
5 = 2( 1ρ)2 1ϒ 1 and h∗1
5 = 1ρ 1.
(64)
Using h∗1
5 from the second equation in the first equations, we
get 1ρ = 12
1∗1
1ϒ
. Substituting this value together with the
second equation of (64) and integrating on y6, it is possible
to compute
h5 = h[0]
5 (x
k, y4)+ 1
4
∫
dy6
( 12)∗1
1ϒ
,
where h[0]
5 (x
k, y4) is an integration function. In result, we
find from the above formula ( 1ρ)2 the coefficient
h6 = 1
4h5
(
∗1
1ϒ
)2
= 1
4
(
∗1
1ϒ
)2 (
h[0]
5 +
1
4
∫
dy6
( 12)∗1
1ϒ
)−1
.
The first part of the N-connection coefficients are found
by integrating two times on y6 in (44) written in the form
1n∗1∗1
i1
= (n∗1
i1
)∗1 = − n∗1
i1
(ln |h5|3/2/|h6|)∗1
for the coefficient 1γ defined in (48). This formula can be
integrated two times on y6 for explicit values of ha1 which
results in
1nk1(x
i , y4, y6) = 11nk1 + 12nk1
∫
dy6
h6
| h5|3/2
= 11nk1 + 12ñk1
∫
dy6
( 1∗1)2
| h5|5/2( 1ϒ)2
= 11nk1 + 12ñk1
∫
dy6
( 1∗1)2
( 1ϒ)2
×
∣∣∣∣h[0]
5 +
1
4
∫
dy6
( 12)∗1
1ϒ
∣∣∣∣
−5/2
,
also containing a second set of integration functions 11nk1
(xi , y4) and redefined second integration function
1
2ñk1
(xi , y4). The second set of N-connection coefficients on
s = 1 can be found from the linear algebraic equations (45)
and express 1wi1 = ∂i1 1/ 1∗1 .
Summarizing the above formulas for the d-connection and
N-connection coefficients on s = 1, we write the data for
generating such generic off-diagonal metrics as stationary
solutions for the system (43)–(45),
h5 = h[0]
5 +
1
4
∫
dy6
( 12)∗1
1ϒ
;
h6 = 1
4
(
∗1
1ϒ
)2 (
h[0]
5 +
1
4
∫
dy6
( 12)∗1
1ϒ
)−1
;
1nk1 = 11nk1 + 12ñk1
∫
dy6
( 1∗1)2
( 1ϒ)2
×
∣∣∣∣h[0]
5 +
1
4
∫
dy6
( 12)∗1
1ϒ
∣∣∣∣
−5/2
;
1wi1 = ∂i1 1/ 1∗1 .
The quadratic elements for off-diagonal 6-d stationary con-
figurations in heterotic supergravity with nonholonomically
induced torsion are constructed using above N-adapted coef-
ficients
ds2K6d = gα1β1(xk, y4, ya1)duα1 duβ1 = ds2K4d [see (59)]
+
[
h[0]
5 +
1
4
∫
dy6
( 12)∗1
1ϒ
]
×
[
dy5 +
(
1
1nk1 + 12ñk1
∫
dy6
( 1∗1)2
( 1ϒ)2
×
∣∣∣∣h[0]
5 +
1
4
∫
dy6
( 12)∗1
1ϒ
∣∣∣∣
−5/2)
dxk1
]
+ 1
4
( 1∗1
1ϒ
)2 (
h[0]
5 +
1
4
∫
dy6
( 12)∗1
1ϒ
)−1
×
[
dy6 + ∂i1
1
1∗1
]
.
(65)
We can repeat the method of constructing solutions for
s = 1 to next shells s = 2 and s = 3 and integrate the sys-
tems (46) and (47), respectively. It is necessary to extend the
formulas for generating and integration functions and effec-
tive sources recurrently as we provided above for redefinition
of respective values for s = 0 to s = 1. The quadratic ele-
ments are parametrized in corresponding forms:
123
17 Page 16 of 32
Eur. Phys. J. C (2017) 77:17
ds2K8d = gα2β2(xk, y4, ya1 , ya2)duα2 duβ2
= ds2K6d [see (65)] +
[
h[0]
6 +
1
4
∫
dy8
( 22)∗2
2ϒ
]
×
[
dy7 +
(
2
1nk2 + 22ñk2
∫
dy8
( 2∗2)2
( 2ϒ)2
×
∣∣∣∣h[0]
7 +
1
4
∫
dy8
( 22)∗2
2ϒ
∣∣∣∣
−5/2)
dxk2
]
+1
4
( 2∗2
2ϒ
)2 (
h[0]
7 +
1
4
∫
dy8
( 22)∗2
2ϒ
)−1
×
[
dy8 + ∂i2
2
2∗2
]
(66)
and
ds2K10d = gα2β2(xk, y4, ya1 , ya2 , ya2)duα3 duβ3
= ds2K8d [see (66)] +
[
h[0]
8 +
1
4
∫
dy10
( 32)∗3
3ϒ
]
×
[
dy9 +
(
3
1nk3 + 32ñk3
∫
dy10
( 3∗3)2
( 3ϒ)2
×
∣∣∣∣h[0]
9 +
1
4
∫
dy10
( 32)∗3
3ϒ
∣∣∣∣
−5/2)
dxk3
]
+ 1
4
( 3∗3
3ϒ
)2 (
h[0]
9 +
1
4
∫
dy10
( 32)∗3
3ϒ
)−1
×
[
dy10 + ∂i3
3
3∗3
]
.
(67)
The generic off-diagonal d-metrics (67) define exact sta-
tionary solutions with Killing symmetries on ∂t and ∂9 in
effective nonholonomic 10-d gravity with sources (34) being
determined by the equations of motion in heterotic super-
gravity (12).
2.3.7 A nonlinear symmetry of generating functions and
effective sources
There is an important nonlinear symmetry relating nontrivial
generating functions and effective sources considered in the
above classes of solutions, used in differen forms in [42,43].
It allows us to redefine the generating functions and introduce
effective cosmological constants instead of effective sources.
Let us study these properties in the case of heterotic string
gravity.
Changing the generating data (, ϒ) ↔ (̃,̃ = const),
where
(2)∗
ϒ
= (̃
2)∗
̃
, which is equivalent to
̃2 =̃
∫
dy4ϒ−1(2)∗ and/or
2 =̃−1
∫
dy4ϒ(̃2)∗,
(68)
up to certain classes of integration functions depending on
coordinates xi , we express the solutions (56) and (57) of the
system of nonlinear PDEs (49) and (51) in two equivalent
forms,
h3 = h3[] = h[0]
3 (x
k)− 1
4
∫
dy4
(2)∗
ϒ
= h3[̃] = h[0]
3 (x
k)−̃
2
4̃
,
and
h4 = h4[] = −
1
4h3[]
(
∗
ϒ
)2
= h4[̃]
= −
1
4h3[̃]
(̃∗)2
̃2
.
Having defined two equivalent nonlinear formulas for ha[]
= ha[̃], we can express in two equivalent forms the N-
adapted coefficients for the d-metric and N-connection (58).
In the second case, the N-connection coefficients are com-
puted using integrals on dy4 for certain values determined
by (̃,̃) via (68) and integration functions,
nk = 1nk + 2nk
∫
dy4
h4[̃]
| h3[̃]|3/2
and
wi = ∂i 
∗
= ∂i 
2
(2)∗
= ∂i
∫
dy4ϒ (̃2)∗
ϒ(̃2)∗
.
We observe that if the effective source ϒ = ϒ(xk) does not
depend on y4, we have the same expression for wi in terms
of generating functions, wi = ∂i 
∗ = ∂ĩ
̃∗ .
The nonlinear symmetry reflects the property of changing
the effective sources mentioned in (34),
ϒ(xk, y4) →  = H+ F+ R,
for φ = 0.
(69)
We can identifỹwith, or any other value H, F, R,
and their sums, depending on the class of models with effec-
tive gauge interactions we consider in our work.
In a similar form, using recurrent formulas, we can prove
the existence of such nonlinear symmetries generalizing (68),
s = 0 : (
2)∗
ϒ
= (̃
2)∗
̃
,
i.e.̃2 =̃
∫
dy4ϒ−1(2)∗
and/or 2 =̃−1
∫
dy4ϒ(̃2)∗;
(70)
s = 1 : (
12)∗1
1ϒ
= (
1̃2)∗1
1̃
,
i.e. 1̃2 = 1̃
∫
dy6( 1ϒ)
−1( 12)∗1
and/or ( 1)2 = 1̃−1
∫
dy6 1ϒ(
1̃2)∗1;
s = 2 : (
22)∗2
2ϒ
= (
2̃2)∗2
2̃
,
123
Eur. Phys. J. C (2017) 77:17
Page 17 of 32
17
i.e. 2̃2 = 2̃
∫
dy8( 2ϒ)
−1( 22)∗2
and/or ( 2)2 = 2̃−1
∫
dy8 2ϒ(
2̃2)∗2;
s = 3 : (
32)∗3
3ϒ
= (
3̃2)∗3
3̃
,
i.e. 3̃2 = 3̃
∫
dy10( 3ϒ)
−1( 32)∗3
and/or ( 3)2 = 3̃−1
∫
dy10 3ϒ(
3̃2)∗3 .
We consider the convention thatϒ̂99 =ϒ̂10
10 = 3ϒ(xk, ya,
ya1 , ya2 , y10) → 3 = φ3 + H3  + F3  + R3  and
identify 3̃ with 3. Such nonlinear transforms can be
used for simplifications of formulas for generic off-diagonal
solutions. Prescribing certain effective matter field initial dis-
tributions, we can redefine the constructions equivalently, via
new classes of generating functions, for effective cosmolog-
ical constants.
Finally, we note that Eq. (67) simplifies for ( 3, 3ϒ) →
( 3̃, 3̃) if 3ϒ = 3ϒ(xi , y4, ya1 , ya2),
ds2K10d = gα2β2(xk, y4, ya1 , ya2 , ya2 , 3̃,3̃)duα3 duβ3
= ds2K8d [see (66)] +
[
h[0]
8 +
( 3̃2)
4 3̃
]
×
[
dy9 +
(
3
1nk3 + 32ñk3
∫
dy10( 3̃∗3)2
×
∣∣∣∣h[0]
9 +
( 3̃2)
4 3̃
∣∣∣∣
−5/2)
dxk3
]
+1
4
( 3̃∗3)2
| 3̃ 3ϒ |
(
h[0]
9 +
( 3̃2)
4 3̃
)−1
×
[
dy10 + ∂i3
3̃
3̃∗3
]
.
Such generic off-diagonal configurations are described by
generic off-diagonal metrics with effective cosmological
constants, generalized generating functions and integration
functions. The contributions of 3ϒ(xi , y4, ya1 , ya2) can be
encoded into generating and integrating functions.
2.3.8 The Levi-Civita conditions
In general, a solution constructed for a generic off-diagonal
metric (24) and canonical d-connections
sD̂ is charac-
terized by nonholonomically induced d-torsion coefficients
T̂γsαsβs (7) completely defined by the N-connection and d-
metric structure. The zero torsion conditions (8) can be sat-
isfied by a subclass of nonholonomic distributions deter-
mined by corresponding parametrizations of the generating
and integration functions and sources. By straightforward
computations (see the details in Refs. [35,38–40]), we can
verify that if the coefficients of N-adapted frames and sv-
components of d-metrics are subject to the respective condi-
tions
s = 0 : w∗
i = ei ln
√| h4|, ei ln√| h3| = 0,
∂iw j = ∂ jwi and n∗
i = 0;
s = 1 : 1w∗1
i1
= 1ei1 ln
√| h6|, 1ei1 ln√| h5| = 0,
∂i1
1w j1 = ∂ j1 1wi1
and
1n∗1
i1
= 0;
s = 2 : 2w∗2
i2
= 2ei2 ln
√| h8|, 2ei2 ln√| h7| = 0,
∂i2
2w j2 = ∂ j2 2wi2
and
2n∗2
i2
= 0;
s = 3 : 3w∗3
i3
= 3ei3 ln
√| h10|, 3ei3 ln√| h9| = 0,
∂i3
3w j3 = ∂ j3 3wi3
and
3n∗3
i3
= 0;
(71)
all d-torsion coefficients are zero.
The n-coefficients solve the conditions (71) if
s = 0 : 2nk(xi ) = 0 and ∂i 1n j (xk) = ∂ j 1ni (xk);
s = 1 : 12nk1(xi1) = 0 and ∂i1 11n j1(xk1) = ∂ j1 11ni1(xk1);
s = 2 : 22nk2(xi2) = 0 and ∂i2 21n j2(xk2) = ∂ j2 21ni2(xk2);
s = 3 : 32nk3(xi3) = 0 and ∂i3 31n j3(xk3) = ∂ j3 31ni3(xk3).
(72)
The explicit form of solutions of constraints on wk derived
from (71) depends on the class of vacuum or non-vacuum
metrics we try to construct; see for details [43]. For instance,
if we choose a generating function =̌(xi , y4), for which
(∂ǐ)
∗ = ∂ǐ∗, we solve the conditions for wi in (71) in
explicit form if ϒ = const, or if such an effective source can
be expressed as a functional, ϒ(xi , y4) = ϒ[̌]. The third
condition for s = 0, ∂iw j = ∂ jwi , can be satisfied for any
generating functionǍ =Ǎ(xk, y4) for which wi =w̌i =
∂ǐ/∂4̌ = ∂iǍ. Following similar considerations for shells
s = 1, 2, 3, we formulate the LC-conditions for generating
functions,
s = 0 :  =̌(xi , y4), (∂ǐ)∗ = ∂ǐ∗,
w̌i = ∂ǐ/∂4̌ = ∂iǍ;
ϒ(xi , y4) = ϒ[̌], or ϒ = const;
s = 1 : 1 = 1̌(uτ , y6), (∂i1 1̌)∗1 = ∂i1 1̌∗1 ,
∂α
1̌/∂6
1̌ = ∂α 1Ǎ; 11nτ = ∂τ 1n(uβ);
ϒ(xi , y4) = ϒ[̌], or ϒ = const;
s = 2 : 2 = 2̌(uτ1 , y8), ∂8∂τ1 2̌ = ∂τ1∂8 2̌;
∂α1
2̌/∂8
2̌ = ∂α2 2Ǎ; 21nτ1 = ∂τ1 2n(uβ1);
ϒ(xi , y4) = ϒ[̌], or ϒ = const;
s = 3 : 2 = 2̌(uτ1 , y8), ∂8∂τ1 2̌ = ∂τ1∂8 2̌;
∂α1; 2̌/∂8 2̌ = ∂α2 2Ǎ; 21nτ1 = ∂τ1 2n(uβ1);
ϒ(xi , y4) = ϒ[̌], or ϒ = const.
(73)
Imposing respective conditions from (72) and (73) on the
coefficients of (58), we define such a class of quadratic ele-
123
17 Page 18 of 32
Eur. Phys. J. C (2017) 77:17
ments for off-diagonal 4-d stationary configurations in het-
erotic supergravity with zero induced torsion,
ds2K4d =ǧαβ(xk, y4)duαduβ = eq [(dx1)2 + (dx2)2]
+
[
h[0]
3 (x
k)− 1
4
∫
dy4
(̌2)∗
ϒ[̌]
]
[dt+(∂k n)dxk]2
− 1
4
(
̌∗
ϒ[̌]
)2 (
h[0]
3 −
1
4
∫
dy4
(̌2)∗
ϒ[̌]
)−1
×[dy4 + (∂iǍ)dxi ]2.
(74)
We can impose similar conditions and generate exact off-
diagonal solutions with respective data ( sǧ, sŇ, s∇̌) for
which all nonholonomic torsions are zero, but this will
impose very strong restrictions on the dynamics of effective
matter fields on extra shells in heterotic supergravity. In order
to consider realistic solutions in string gravity with 6-d inte-
rior almost-K ähler configurations, the torsion is positively
not zero. Such 10-d solutions are conventionally parame-
terized as (s = 0 :ǧ, N ,∇; s = 1, 2, 3 :
sg, sN, sD̂),
for stationary configurations with Killing symmetry on ∂t
and ∂9. Hereafter we shall work with solutions with nontriv-
ial nonholonomically induced torsions considering that it is
always possible to state additional constraints resulting in
LC-configurations in the 4-d case or extra dimensions.
2.4 Small N-adapted nonholonomic stationary
deformations
We can construct very general classes of generic off-diagonal
stationary solutions in heterotic supergravity. It is not clear
what physical meaning these configurations may have for
generating and integration functions with arbitrary data.
Using small off-diagonal deformations of some known phys-
ically important solutions we can understand physical prop-
erties of such solutions characterized by locally anisotropic
polarization/running of constants and nonlinear off-diagonal
gravitational interactions determined by (super-) string cor-
rections.
We consider a “prime” pseudo-Riemannian metric of type
g̊ = [g̊i ,h̊as ,N̊ js
bs
] when
ds2 =g̊i (xk)(dxi )2 +h̊a(xk, y4)(dya)2(e̊a)2
+g̊a1(xk, y4, y6)
(
e̊a1
)2
+g̊a2(xk, y4, ya1 , y8)
(
e̊a2
)2
+g̊a3(xk, y4, ya1 , ya2 , y10)
(
e̊a3
)2
,
(75)
wheree̊as are N-elongated differentials. Such a metric is
diagonalizable if there is a coordinate transform uα
′
s =
uα
′
s (uαs ) when ds2 =
g̊i ′(xk′)(dxi
′
)2 +h̊a′s (xk′)(dya
′
s )2,
with sẘis =
sn̊is = 0. To construct exact solutions with
non-singular coordinates it is important to work with “for-
mal” off-diagonal parameterizations when the coefficients
sẘis and/or
sn̊is are not zero but the anholonomy coeffi-
cientsW̊αs
βsγs
(uμs ) = 0; see (4). We suppose that some data
(g̊i ′,h̊a′s ) may define a known physically important diagonal
exact solution in GR or heterotic string gravity (for instance, a
black hole, BH, configuration of Kerr or Schwarzschild type).
Our goal is to study certain small generic off-diagonal para-
metric deformations of the prime d-metric and N-connection
coefficients (75) into certain target metrics
ds2 = ηig̊i (dxi )2 + ηasg̊as (eas )2,
(76)
e3 = dt + nηin̊idxi , e4 = dy4 + wηiẘidxi ,
e5 = dy5 + nηi1n̊i1 dxi1 , e6 = dy6 + wηi1ẘi1 dxi1 ,
e7 = dy7 + nηi2n̊i2 dxi2 , e8 = dy8 + wηi2ẘisdxis ,
e9 = dy9 + nηi3n̊i3 dxi3 , e10 = dy10 + wηi3ẘi3dxi3 ,
where the coefficients (gαs = ηαsg̊αs ,w ηisẘis , nηis nis )
define, for instance, a d-metric sg (24) as a solution of non-
holonomic equations of motion in heterotic string gravity
(12) rewritten as a nonlinear system of PDEs in nonholo-
nomic 10-d gravity (39)–(47).
Let us construct exact solutions, for instance, of type (67).
For certain well-defined conditions, we can show an expres-
sion using d-metric and N-connection deformations stated
in explicit form on all shells for the d-metric and n- and w-
coefficients:
For the coefficients of d-metrics,
(77)
ηi = 1 + εχi (xk), ηa = 1 + εχa(xk, y4),
ηa1 = 1 + εχa1(xk, y4, y6),
ηa2 = 1 + εχa2(xk, y4, ya1 , y8),
ηa3 = 1 + εχa3(xk, y4, ya1 , ya2 , y10);
and for the coefficients of N-connection,
nηi = 1 + ε nηi (xk, y4),
wηi = 1 + ε wχi (xk, y4),
nηi1 = 1 + ε nηi1(xk, y4, y6),
wηi1 = 1 + ε wχi1(xk, y4, y6),
nηi2 = 1 + ε nηi2(xk, y4, ya1 , y8),
wηi2 = 1 + ε wχi2(xk, y4, ya1 , y8),
nηi3 = 1 + ε nηi3(xk, y4, ya1 , ya2 , y10),
wηi3 = 1 + ε wχi3(xk, y4, ya1 , ya2 , y10),
for a small parameter 0 ≤ ε  1, when (76) transforms
into (75) for ε → 0 (which in turn, can be diagonalized).
In general, there are no smooth limits from such nonholo-
nomic deformations which can be satisfied for arbitrary gen-
eration and integration functions, integration constants, and
general (effective) sources on the corresponding shells. The
goal of this subsection is to analyze such conditions when ε-
deformations with nontrivial N-connection coefficients for
123
Eur. Phys. J. C (2017) 77:17
Page 19 of 32
17
the prime and target d-metrics can be related to new classes
of solutions of motion heterotic string equations.
We denote nonholonomic ε-deformations of certain prime
d-metric (75) into a target one (76) with polarizations (77) in
the formg̊ → εg = ( εgi , εhas , εNas−1
bs
). The goal of this
subsection is to compute the formulas for ε-deformations of
prime d-metrics resulting in stationary solutions in equivalent
nonholonomic 10-d gravity with Killing symmetries on ∂t
and ∂9.
The geometric constructions will be provided in detail
for the 4-d configurations and then extended to higher-
dimensional shells. Deformations of h-components are char-
acterized by εgi =g̊i (1 + εχi ) = eq(xk ) resulting in a solu-
tion of the 2-d Laplace equation (39). For q =
0q(xk) +
ε 1q(xk) and hϒ = 0hϒ(xk) +
1̃
hϒ(x
k), we compute the
deformation polarization functions χi = e 0q 1q/g̊i 0hϒ. In
this formula, we use certain generating and source functions
as solutions of 0q•• + 0q ′′ = 0ϒ and 1q•• + 1q ′′ = 1ϒ .
At the next step, we compute ε-deformations of v-
components on the s = 0 shell,
εh3 = h[0]
3 (x
k)− 1
4
∫
dy4
(2)∗
ϒ
= (1 + εχ3)g̊3,
(78)
εh4 = −1
4
( ∗)2
ϒ2
(
h[0]
3 −
1
4
∫
dy4
(2)∗
ϒ
)−1
= (1 + εχ4)g̊4;
(79)
Parametrizing the generation function,
 → ε =̊(xk, y4)[1 + εχ(xk, y4)],
(80)
and introducing this value in (79), one obtains
χ3 = − 1
4g̊3
∫
dy4
(̊2χ)∗
ϒ
and
∫
dy4
(̊2)∗
ϒ
= 4(h[0]
3 −g̊3).
(81)
In result, we can compute χ3 for any deformation χ from a 2-
hypersurface y4 = y4(xk). Such a hypersurface, in general,
is defined in non-explicit form from̊ =̊(xk, y4) when
the integration function h[0]
3 (x
k), the prime valueg̊3(xk), and
the fraction (̊2)∗/ϒ satisfy the condition (81).
We can find the formula for hypersurface̊(xk, y4) pre-
scribing a value of ϒ. Introducing (64) into (79 ), one obtains
χ4 = 2
(
χ +̊
̊∗
χ∗
)
− χ3
= 2
(
χ +̊
̊∗
χ∗
)
+ 1
4g̊3
∫
dy4
(̊2χ)∗
ϒ
,
i.e. we compute χ4 for any data (̊,g̊3,χ).The formula for a
compatible source is hϒ = ±̊∗/2
√
|g̊4h[0]
3 |. It transforms
(81) into a 2-d hypersurface formula y4 = y4(xk) defined in
non-explicit form from
∫
dy4̊ = ±(h[0]
3 −g̊3)/
√
|g̊4h[0]
3 |.
(82)
The ε-deformations of N-connection coefficients wi =
∂i/
∗ for nontrivialẘi = ∂i̊/̊∗ are found following
Eqs. (64) and (77), wχi = ∂i (χ̊)
∂i̊
− (χ̊)∗
̊∗
, where there is
no summation on index i. We can compute the deformations
of the n-coefficients (we omit such details; see below for the
necessary formulas).
In a similar way, the ε-deformations of 1v-components
on the s = 1 shell are computed,
εh5 = h[0]
5 (x
k, y4)+
∫
dy6
( 12)∗1
4 1ϒ
= (1 + εχ5)g̊5,
εh6 = − (
1∗1)2
4 1ϒ2
(
h[0]
5 +
1
4
∫
dy6
( 12)∗1
1ϒ
)−1
= (1 + εχ6)g̊6.
The first shell generation function is parameterized 1 →
1
ε = 1̊(xk, y4, y6)[1 + ε 1χ(xk, y4, y6)], resulting in
χ5 = − 1
4g̊5
∫
dy6
( 1̊2 1χ)∗1
1ϒ
and
∫
dy6
( 1̊2)∗1
1ϒ
= 4(h[0]
5 −g̊5).
It is possible to compute χ5 for any deformation 1χ from
a 3-hypersurface y6 = y6(xk, y4) (for stationary solutions
with Killing symmetries on ∂t and ∂5). We note that, in gen-
eral, such a hypersurface is defined in non-explicit form
from 1̊ =
1̊(xk, y4, y6) when the integration func-
tion h[0]
5 (x
k, y4), the prime valueg̊5(xk, y4) and the fraction
( 1̊2)∗1/ 1ϒ satisfy a condition similar to (81).
We can find the formula for hypersurface 1̊(xk, y4, y6)
by prescribing a first shell value of effective source 1ϒ.
Similarly to (64) and (79), it is possible to generalize and
compute
χ6 = 2
(
1χ +
1̊
1̊∗1
1χ∗1
)
− χ5
= 2
(
1χ +
1̊
1̊∗1
1χ∗1
)
+ 1
4g̊5
∫
dy6
( 1̊2 1χ)∗1
1ϒ
,
i.e. we compute χ6 for any data ( 1̊,g̊5,1 χ). One defines a
first shell compatible source if 1ϒ = ± 1̊∗1/2
√
|g̊6h[0]
5 |.
It generalizes (81) into a 2-d hypersurface formula y6 =
y6(xk, y5) which has to be computed in non-explicit form
from
∫
dy6 1̊ = ±(h[0]
5 −g̊5)/
√
|g̊6h[0]
5 |. On the first
shell, the ε-deformations of N-connection coefficients wi1 =
∂i1
1/ 1∗1 for nontrivial
ẘi1 = ∂i1 1̊/ 1̊∗1 are
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17 Page 20 of 32
Eur. Phys. J. C (2017) 77:17
wχi1 = ∂i1 (
1χ 1̊)
∂i1
1̊
− ( 1χ 1̊)∗1
1̊∗1
, where there is no sum-
mation on index i1. We omit computations of deformations
of the n-coefficients but we shall present necessary formulas
below, see similar details in [43].
Summarizing the above formulas and extending for all
shells s = 0, 1, 2, 3, we obtain such coefficients for ε-
deformations of a prime metric (75) into a target stationary
metric satisfying the equations of motion in heterotic super-
gravity:
εgi =g̊i [1 + εχi ] = [1 + εe 0q 1q/g̊i 0hϒ]g̊i
as a solution of 2-d Poisson equations (39);
εh3 = [1 + ε χ3]g̊3 =
[
1 − ε 1
4g̊3
∫
dy4
(̊2χ)∗
ϒ
]
g̊3;
εh4 = [1 + ε χ4]g̊4 =
[
1 + ε
(
2
(
χ +̊
̊∗
χ∗
)
+ 1
4g̊3
∫
dy4
(̊2χ)∗
ϒ
)]
g̊4;
(83)
εni = [1 + ε nχi ]n̊i =
[
1 + εñi
∫
dy4
1
ϒ2
×
(
χ +̊
̊∗
χ∗ + 5
8g̊3
(̊2χ)∗
ϒ
)]
n̊i ;
εwi = [1 + ε wχi ]ẘi =
[
1 + ε
(
∂i (χ̊)
∂i̊
− (χ̊)
∗
̊∗
)]
ẘi ,
whereñi (xk) is a redefined integration function including
contributions from the prime metric. On a shell s, these for-
mulas are defined recurrently (we omit parameterizations of
functions on shell coordinates because they can be deter-
mined in compatible form with indices and labels for shells),
εh3+2s = [1 + ε χ3+2s]g̊3+2s
=
[
1 − ε
1
4g̊3+2s
∫
dy4+2s (
s̊2 sχ)∗s
sϒ
]
g̊3+2s;
εh4+2s = [1 + ε χ4+2s]g̊4+2s
=
[
1 + ε
(
2
(
sχ +
s̊
s̊∗s
sχ∗s
)
+
1
4g̊3+2s
∫
dy4+2s (
s̊2 sχ)∗1
sϒ
)]
g̊4+2s;
εnis = [1 + ε nχis ]n̊is
=
[
1 + εñis
∫
dy5+2s
1
sϒ2
×
(
sχ +
s̊
s̊∗
sχ∗s+
5
8g̊3+2s
( s̊2 sχ)∗s
sϒ
)]
n̊is ;
εwis = [1+ε wχis ]ẘis
=
[
1 + ε
(
∂is (
sχ s̊)
∂is
s̊
− (
sχ s̊)∗s
s̊∗s
)]
ẘis .
These formulas with s = 1, 2, 3 allow us to parametrize
all coefficients of vertical components of d-metrics and N-
connections. For small parametric deformations, the values
χ, sχ and ϒ, sϒ have to be considered as generating func-
tions. The values with a “circle” are prescribed by a chosen
prime solution (in our case, we can chose the 4-d Kerr met-
ric embedded into a 10-d gravity string spacetime). Fixing a
small value ε, we compute such deformations and prove their
stability (see [38] and the references therein) for any stable
prime solution.
The ε-deformed quadratic elements are written
ds2εt = εgαsβs (xk, y4, ya1 , ya2 , y10)duαsduβs
= εgi
(
xk
)
[(dx1)2 + (dx2)2] + εh3(xk, y4)[dt
+ εnk (xk, y4)dxk]2 + εh4(xk, y4)
×[dy4 + εwi (xk, y4)dxi ]2 + εh5(xk, y4, y6)
×[dy5 + εnk1 (xk, y4, y6)dxk1 ]2
+ εh6(xk, y4, y6) [dy6 + εwi1(xk, y4, y6)dxi1 ]2
+ εh7(xk, y4, ya1 , y8)[dy7
+ εnk2 (xk, y4, ya1 , y8)dxk2 ]2
+ εh8(xk, y4, ya1 , y8) [dy8
+ εwi2(xk, y4, ya1 , y8)dxi2
+ εh9(xk, y4, ya1 , ya2 , y10)
×[dy9 + εnk3 (xk, y4, ya1 , ya2 , y10)dxk3 ]2
+ εh10(xk, y4, ya1 , ya2 , y10)
×[dy10 + εwi3(xk, y4, ya1 , ya2 , y10)dxi3 ].
(84)
We can impose additional constraints in order to extract LC-
configurations as we considered in Sect. 2.3.8.
3 Nonholonomic heterotic string deformations of the
Kerr metric
In this section, we study generic off-diagonal deformations
and generalizations of the 4-d Kerr metric to new classes
of exact solutions of equations of motion in heterotic string
theory; see [31,32,52,71]. We prove that using the AFDM
extended to models with almost-Kähler internal spaces, the
Kerr solution can be constructed as a particular case by pre-
scribing a corresponding class of generating and integration
functions. In general, such solutions are with nontrivial cos-
mological constants and nonholonomically induced torsions.
Imposing additional nonholonomic constraints, we can gen-
erate effective vacuum solutions and extract Levi-Civita con-
figurations. A series of new classes of small parametric solu-
123
Eur. Phys. J. C (2017) 77:17
Page 21 of 32
17
tions when the Kerr metrics are nonholonomically deformed
into general or ellipsoidal stationary configurations in four-
dimensional gravity and/or extra dimensions are considered.
We provide and study examples of generic off-diagonal met-
rics encoding nonlinear interactions with 3-form and gauge-
like fields, nonholonomically induced torsion effects and
instanton configurations.
3.1 Preliminaries on the Kerr vacuum solution and
nonholonomic variables
A 4-d ansatz
ds2
[0] = Y−1e2h(dρ2 + dz2)− ρ2Y−1dt2 + Y (dϕ + Adt)2
written in terms of three functions (h,Y, A) on coordinates
xi = (ρ, z) defines the Kerr solution of the vacuum Einstein
equations (for rotating black holes) if we choose
Y = 1 − (px̂1)
2 − (qx̂2)2
(1 + px̂1)2+(qx̂2)2 , A=2M
q
p
(1 −x̂2)(1 + px̂1)
1 − (px̂1)− (qx̂2) ,
e2h = 1 − (px̂1)
2 − (qx̂2)2
p2[(̂x1)2 + (̂x2)2] , ρ
2 = M2(̂x21 − 1)(1 −x̂22 ),
z = Mx̂1x̂2.
Some values M = const and ρ = 0 result in a horizon
x̂1 = 0 and the “north/south” segments of the rotation axis,
x̂2 = +1/ − 1. For further applications of the AFDM, we
can write this prime solution in the form
ds2
[0] = (dx1)2 + (dx2)2 − ρ2Y−1(e3)2 + Y (e4)2.
(85)
This is possible if the coordinates x1(̂x1,x̂2) and x2(̂x1,x̂2)
are defined for any
(dx1)2 + (dx2)2
= M2e2h (̂x21 −x̂22 )Y−1
(
dx̂21
x̂21 − 1
+ dx̂
2
2
1 −x̂22
)
and y3 = t +ŷ3(x1, x2), y4 = ϕ +ŷ4(x1, x2, t). We can
consider an N-adapted basis e3 = dt + (∂iŷ3)dxi , e4 =
dy4 + (∂iŷ4)dxi , for some functionsŷa, a = 3, 4, with
∂tŷ4 = −A(xk).
The Kerr metric was intensively studied in the so-called
Boyer–Lindquist coordinates (r,ϑ,ϕ, t), for r = m0(1 +
px̂1),x̂2 = cosϑ, which can be considered for applications
of the AFDM. Such coordinates are expressed via parameters
p, q which are related to the total black hole mass, m0 and
the total angular momentum, am0, for the asymptotically flat,
stationary, and antisymmetric Kerr spacetime. The formulas
m0 = Mp−1 and a = Mqp−1 when p2 + q2 = 1 imply
m20 − a2 = M2. In these variables, the metric (85) can be
written
ds2
[0] = (dx1
′
)2 + (dx2′)2 + A(e3′)2 + (C − B2/A)(e4′)2,
e3
′ = dt + dϕB/A = dy3′ − ∂i ′(ŷ3′ + ϕB/A)dxi ′ ,
e4
′ = dy4′ = dϕ,
(86)
for any coordinate functions x1
′
(r,ϑ), x2
′
(r,ϑ), y3
′ =
t+ŷ3′(r,ϑ,ϕ)+ϕB/A, y4′ = ϕ, ∂ϕŷ3′ = −B/A, for which
(dx1
′
)2+(dx2′)2 =  (−1dr2 + dϑ2), and the coefficients
are
A = −−1(− a2 sin2 ϑ),
B = −1a sin2 ϑ[− (r2 + a2)],
C = −1 sin2 ϑ[(r2 + a2)2 −a2 sin2 ϑ], and
 = r2 − 2m0 + a2,  = r2 + a2 cos2 ϑ.
(87)
We refer the reader to [31,52,71] for main results and meth-
ods for stationary black hole solutions (the coordinatesx̂1,x̂2
introduced above correspond to x, y, respectively, in chapter
4 of the first book).
The primed quadratic linear elements (85) (or (86))
g̊1 = 1,g̊2 = 1,h̊3 = −ρ2Y−1,h̊4 = Y,N̊ a
i = ∂iŷa, or
(88)
g̊1′ = 1,g̊2′ = 1,h̊3′ = A,h̊4′ = C − B2/A,
N̊ 3
i ′ =n̊i ′ = −∂i ′(ŷ3
′ + ϕB/A),N̊ 4
i ′ =ẘi ′ = 0
define solutions of vacuum Einstein equations parametrized
in the form (16) and (17) with zero sources. A straightfor-
ward application of the AFDM is possible if we consider
a correspondingly N-adapted system of coordinates instead
of the “standard” prolate spherical, or Boyer–Lindquist sys-
tem. Parametrizations (88) are most convenient for a straight-
forward application of the AFDM. This way we can gener-
alize the solutions for coefficients depending on more than
two coordinates, with non-Killing configurations and/or extra
dimensions.
Working with general classes of stationary solutions gen-
erated by the AFDM, the Kerr vacuum solution in GR can be
considered as a “degenerate” case of 4-d off-diagonal vacuum
solutions determined by primary metrics with data (88) when
the diagonal coefficients depend only on two “horizontal”
N-adapted coordinates. Such a metric contains off-diagonal
terms induced by rotation frames in a form when the non-
holonomically induced torsion is zero. In N-adapted frames,
further generic off-diagonal and extra dimension generaliza-
tions can be performed following standard geometric meth-
ods (see the following sections).
3.2 Off-diagonal deformations of 4-d Kerr metrics
by heterotic string sources
Let us consider the coefficients (88) for the Kerr metric as the
prime metricg̊ (in general, a prime metric may or may not
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17 Page 22 of 32
Eur. Phys. J. C (2017) 77:17
be an exact solution of the Einstein or other modified grav-
itational equations). Our goal is to construct nonholonomic
deformations,
(g̊,N̊,
v
ϒ̊ = 0,ϒ̊ = 0) → (g,N, hϒ(xk) → h 
= const
= 0,ϒ →  = const
= 0);
see the sources (34), (37), and (19), respectively, with (21),
when hϒ(xk) → h = Hh + Fh + Rh , ϒ(xk, y4) →
 = H + F + R and φh = φ = 0, as in (69).
The main condition is that the target metric g is of type (74),
which positively defines a torsionless off-diagonal solution
of field equations in the 4-d gravity sector with sources deter-
mined from the heterotic string theory (12) with ansatz for
the sources (38). The N-adapted deformations of coefficients
of metrics, frames, and sources are parametrized in the form
[g̊i ,h̊a,ẘi ,n̊i ] → [gi = ηig̊i , h3 = η3h̊3, h4
= η4h̊4,wi =ẘi + ηwi , ni =n̊i + ηni ,ẘi = 0],
hϒ = ϒ = K = −5[( Hs)2 + nF ( Fs)2( Rs)2],̌2
= exp[̌(xk′ , y4)],h̊3 = h(0)
3 ,h̊4 = −ϒ̌2/ K2
(89)
The primesg̊i ,h̊a,ẘi ,n̊i (88) are given by coefficients
depending only on (xk′). The general deformations of the
Kerr solution determined by generating function and extra-
dimensional string sources with ansatz resulting in cosmo-
logical constants described in terms of polarization functions,
where |η4′ |, may not have a smooth limit to 1,
η3′ = 1 − (̌2)/ Kh̊3, η4′ = (̌∗)2/( K)2η3′h̊3,
ηwi ′ = ∂i ′Ǎ, ηnk′ = ∂k′ ηn(xi ′),
where the coefficient 1/4 was introduced iň and the func-
tionǍ has any value for which ∂i ′̌/̌∗ = ∂i ′Ǎ.
Summarizing the above formulas, we obtain the quadratic
element
ds2Kη4d =ǧαβ(xk , y4)duαduβ = ηig̊i (dxi )2 + ηag̊a(ea)2
= eq[ K][(dx1′)2 + (dx2′)2] +
(
A −̌
2
K
)
×
[
dy3
′ +
(
∂k′
ηn(xi
′
)− ∂k′
(
ŷ3
′ + ϕ B
A
))
dxk
′
]2
+
(̌∗)2
K
(
KA −̌2
)
(
C − B
2
A
)
[dϕ + ∂i ′Ǎ)dxi
′ ]2,
(90)
where wi ′ =ẘi ′ + ηwi ′ = ∂i ′( ηÃ[ ]), nk′ =n̊k′ + ηnk′ =
∂k′(−ŷ3′ + ϕB/A + ηn) and q is a solution of
q•• + q ′′ = 2 K.
We used the important relationh̊3′h̊4′ = AC − B2 and
emphasize that it is possible to take any function ηn(xk).
The solutions (90) are for stationary LC-configurations
determined by off-diagonal heterotic string gravity effects
on Kerr black holes when the new class of spacetimes come
with Killing symmetry on ∂/∂y3
′
and generic dependence
on three (from maximally four) coordinates, (xi
′
(r,ϑ), ϕ).
Similar solutions were constructed and studied in massive
gravity with extra dimensions [43]. Off-diagonal modifica-
tions are possible for any nontrivial values of̌ and any
small constants H, F, R. The solutions depend on the
type of generating functioň(xi
′
,ϕ) we fix in order to suit
certain experimental/observational data for fixed systems of
reference/coordinates. These can be re-parameterized for an
effective K, which should also be compatible with experi-
mental data. In such variables, we can mimic stationary het-
erotic string gravity effects by off-diagonal configurations in
GR with integration parameters which should also be fixed
by additional assumptions on symmetries of interactions. See
Sect. 3.3 for ellipsoid configurations and details on paramet-
ric Killing symmetries in Refs. [33–35].
3.2.1 Nonholonomically string induced torsion for Kerr
metrics in the 4-d sector
If we do not impose the LC-conditions (17), a nontrivial
source ϒ →  from heterotic string gravity induces sta-
tionary configurations with nontrivial d-torsion (7). The tor-
sion coefficients are determined by metrics of type (59) with
hϒ = ϒ = K as in (89) and parametrizations of coef-
ficients and coordinates distinguishing the prime data for a
Kerr metric (88). Such solutions can be written in the form
ds2 = eq [(dx1′)2 + (dx2′)2]
+
(
A − 
2
K
)[
dy3
′ +
(
1nk′(x
i ′)
+ 2nk′(xi ′)
∫
dy4(2)∗(KA −2)−5/2
−∂k′
(
ŷ3
′ + ϕ B
A
))
dxk
′
]2
− (
∗)2
K
AC − B2
KA −2
[
dϕ + ∂i ′
∂ϕ
dxi
′
]2
,
(91)
where the generating function (xi
′
,ϕ) is not subject to any
integrability conditions. Nontrivial stationary off-diagonal
torsion effects may result in additional effective rotations if
the integration function 2nk
= 0. Considering two different
classes of off-diagonal solutions (91) and (90), we can study
if heterotic string corrections to GR can have a nonholonom-
ically induced torsion or if such effects are characterized by
additional nonholonomic constraints as in GR (for zero tor-
sion).
It should be noted that configurations of type (91) can
be constructed in various theories with noncommutative and
123
Eur. Phys. J. C (2017) 77:17
Page 23 of 32
17
commutative variables. We can consider warped and trapped
brane type variables in string, Finsler-like and/or Hořava–
Lifshitz theories [24,38,51,55,56] when nonholonomically
induced torsion effects play a substantial role.
3.2.2 Small modifications of Kerr metrics and effective
string sources
It is not clear what physical meaning general deformations
of the Kerr metric described by metrics of type (91) may
have. We can choose certain subclasses of nonholonomic
distributions describing stationary ε-deformations described
by Eqs. (83). Using the Kerr solution (88) as a primary metric
with assumptions for the string sources, we compute small
deformations into d-metric and N-connection coefficients,
ds2K ε4d =ǧαβ(xk, y4)duαduβ = [1 + εχi (xk)]g̊i (dxi )2
+[1 + εχa(xk, y4)]g̊a(ea)2.
The ε-deformations are computed thus:
εgi =g̊i [1 + εχi ] = [1 + εe 0q 1q/K]g̊i
as a solution of 2-d Poisson equations (39);
εh3 = [1 + ε χ3]g̊3 =
[
1 − ε̊
2χ
4A K
]
g̊3;
εh4 = [1 + ε χ4]g̊4
=
[
1 + ε
(
2
(
χ +̊
̊∗
χ∗
)
+̊
2χ
4A K
)]
g̊4;
εni = [1 + ε nχi ]n̊i
=
[
1 + εñi
∫
dy4
(
χ +̊
̊∗
χ∗ + 5(̊
2χ)∗
8AK
)]
n̊i ;
εwi = [1 + ε wχi ]ẘi =
[
1 + ε
(
∂i (χ̊)
∂i̊
− (χ̊)
∗
̊∗
)]
ẘi ,
whereg̊1′ = 1,g̊2′ = 1,h̊3′ = A,h̊4′ = C − B2/A,N̊ 3
i ′ =
n̊i ′ = −∂i ′(ŷ3′ + ϕB/A) are kept as above but a coordinate
transform is performed in order to haveN̊ 4
i ′ =ẘi ′
= 0. We
use polarization functions linearized on ε,
ηi = 1 + εχi (xk), ηa = 1 + εχa(xk, y4);
and for the coefficients of N-connection,
nηi = 1 + ε nηi (xk, y4), wηi = 1 + ε wχi (xk, y4).
Summarizing the above for the 4-d ε-configurations, we
obtain the quadratic element
ds2K ε4d =
(
1 + εe 0q
1q
K
)
[(dx1′)2 + (dx2′)2]
+
(
A − ε̊
2χ
4 K
)[
dy3
′ −
[
1 + εñi
∫
dy4
×
(
χ +̊
̊∗
χ∗ + 5(̊
2χ)∗
8AK
)]
∂k′
(
ŷ3
′ + ϕ B
A
)
dxk
′
]2
+
[
1 + ε
(
2(χ +̊
̊∗
χ∗)+̊
2χ
4A K
)]
×
(
C − B
2
A
)[
dϕ+
[
1+ε
(
∂i (χ̊)
∂i̊
− (χ̊)
∗
̊∗
)]
ẘi
]2
.
In general, these heterotic string deformations of the Kerr
metric are nonholonomically induced torsion coefficients,
linear in ε.
We can impose additional constraints on χ,ñi and̊
which allows us to extract LC-configurations. The corre-
sponding ε-deformed analog of the metric (90) can be written
ds2Kη4d =
(
1 + εe 0q
1q
K
)
[(dx1′)2 + (dx2′)2]
+
(
A − ε̊
2χ
4 K
)[
dy3
′ +
(
ε∂k′
χn(xi
′
)
−∂k′
(
ŷ3
′ + ϕ B
A
))
dxk
′
]2
+
[
1 + ε
(
2
(
χ +̊
̊∗
χ∗
)
+̊
2χ
4A K
)]
×
(
C − B
2
A
)
[dϕ + ε∂i ′Ǎ)dxi ′ ]2,
(92)
where χ defines a deformed generating function
ε =
̊(xk, y4)[1 + εχ(xk, y4)] as in Eq. (80) but subjected to
the condition
ε∂i ′Ǎ = ∂i ′( ε)/( ε)∗,
which together with χn(xi
′
) are chosen to result in zero
nonholonomically induced torsion.
3.3 String induced ellipsoidal 4-d deformations of the Kerr
metric
We provide some examples of how the Kerr primary data (88)
are nonholonomically deformed by heterotic string interac-
tions into target generic off-diagonal solutions of vacuum
and non-vacuum Einstein equations for the canonical d-
connection and/or the Levi-Civita connection. Generic off-
diagonal metrics of type (92) can be parameterized as ellip-
soidal deformations of the Kerr metric on a small eccentricity
parameter ε.
3.3.1 Ellipsoidal configurations with string induced
cosmological constant
Let us construct solutions for ε-deformations of type (92)
with ellipsoidal configurations. We choose a generating func-
tion χ3′, when the constraint h3′ = 0 defines a stationary
123
17 Page 24 of 32
Eur. Phys. J. C (2017) 77:17
rotoid configuration (different from the ergo sphere for the
Kerr solutions): We prescribe
χ3′ =
̊2χ
4A K
= 2ζ sin(ω0ϕ + ϕ0),
(93)
for constant parameters ζ ,ω0, and ϕ0, where the values
A(r,ϑ)[1 + εχ3′(r,ϑ,ϕ)] =Â(r,ϑ,ϕ)
= −−1(̂−a2 sin2 ϑ) and̂(r,ϕ)=r2−2m(ϕ)+a2,
are considered as ε-deformations of Kerr coefficients (87).
We get an effective “anisotropically polarized” mass
m(ϕ) = m0/(1 + εζ sin(ω0ϕ + ϕ0)).
(94)
As regards the result, the conditionh3 = 0, i.e. ϕ(r,ϕ,ε) =
a2 sin2 ϑ, states an ellipsoidal “deformed horizon”
r(ϑ, ϕ) = m(ϕ)+ (m2(ϕ)− a2 sin2 ϑ)1/2.
If a = 0, we obtain the parametric formula for an ellipse with
eccentricity ε, r+ =
2m0
1+εζ sin(ω0ϕ+ϕ0) . Such configurations
correspond to the generating function
χ = 8̃ζ A K
̊2
sin(ω0ϕ + ϕ0)
(95)
determined by effective heterotic string source K, as fol-
lows from (93).
If the anholonomy coefficients (4) computed for (92) are
not trivial for wi and nk = 1nk, the generated solutions
cannot be diagonalized via coordinate transforms.
The corresponding nonholonomically deformed 4-d space-
times have one Killing symmetry on ∂/∂y3
′
. For small ε, the
singularity at  = 0 is “hidden” under ellipsoidal deformed
horizons if m0 ≥ a. Similarly to the Kerr solution, there are
ϕ-deformed cases for both the event horizon,
r+ = m(ϕ)+ (m2(ϕ)− a2 sin2 ϑ)1/2,
and the Cauchy horizon,
r− = m(ϕ)− (m2(ϕ)− a2 sin2 ϑ)1/2,
which are effectively embedded into an off-diagonal back-
ground determined by N-coefficients. Using an ellipsoid type
generating function (95) in (92), we construct a class of
generic off-diagonal solutions of effective Einstein equa-
tions with heterotic string gravity effective cosmological
constant K, which in its turn can be related to arbitrary
sources via a redefinition of the generating functions (see
Eqs. (70) and (37)) adapted to ε-deformations. The corre-
sponding quadratic line elements are
ds2Kη4d =
(
1 + εe 0q
1q
K
)
[(dx1′)2 + (dx2′)2] + A[1
−2εζ̃ sin(ω0ϕ + ϕ0)]
[
dy3
′
+
(
ε∂k′
χn(xi
′
)− ∂k′
(
ŷ3
′ + ϕ B
A
))
dxk
′
]2
+
[
1 + ε
(
(8A K+̊2)̃ζ
4 K̊2
sin(ω0ϕ + ϕ0)
−16̃ζω0A K
3̊2
cos(ω0ϕ + ϕ0)
)]
×
(
C − B
2
A
)
[dϕ + ε∂i ′Ǎ)dxi ′ ]2.
(96)
The new classes of ε-deformed solutions determine Kerr-
like black hole solutions with additional dependencies on
the variable ϕ of certain diagonal and off-diagonal coeffi-
cients of the metric. There is an obvious anisotropy in the
angle ϕ. The valuesζ̃ and ω0 have to be chosen in accor-
dance with experimental data. The function̊ depends on
corresponding frame distributions for the prime metric. Fix-
ing a = 0 for a ε
= 0, we get ellipsoidal deformations of
the Schwarzschild black holes. We studied these construc-
tions in detail in [38]; see also the references therein on
the stability and interpretation of such solutions with both
commutative and/or noncommutative deformation parame-
ters. In general, a black hole/ellipsoid interpretation is not
possible for “non-small” N -deformations of the Kerr metric.
For certain embeddings, we can generate black hole-like con-
figurations with deformed horizons and locally anisotropic
polarized physical constants.
3.3.2 Ellipsoid Kerr–de Sitter configurations in R2 and
heterotic string gravity
The first examples of generic off-diagonal ellipsoid–solitonic
deformations of similar Kerr–Sen black holes were con-
structed in [72]. Recently, asymptotically de Sitter solutions
with spherical symmetry for R2 gravity were studied in [14]
and the nonholonomic geometric off-diagonal evolution of
such metrics was analyzed in [70]. In this section, we show
that those constructions can be related to the 4-d part of het-
erotic string MGTs.
We consider a prime 4-d metric
ds2 = 3λ
2ς2
{(
1 − M
r
− λr2
)−1
dr2 + r2dθ2
+ r2 sin θdϕ2 −
(
1 − M
r
− λr2
)
dt2
}
,
(97)
which for
e
√
1/3φ = 3λ
2ς2
= 1
8ς2
R and
gμν = e
√
1/3φgμν = R
8ς2
gμν, R
= 0,
123
Eur. Phys. J. C (2017) 77:17
Page 25 of 32
17
defines exact solutions with spherical symmetry in R2 grav-
ity, for equations Rμν = 2ς2gμν. The effective cosmo-
logical constant ς2 is usually related to nonlinear scalar
fields/dilaton-like interactions in effective Einstein gravity
resulting from R2 gravity. In our model of heterotic string
gravity, we can choose
2ς2 = K
(98)
and study quadratic gravity 4-d models determined by het-
erotic string effective sources. The metric (97) describes
asymptotically de Sitter solutions with λ > 0 and R
= 0.
Introducing new 4-d coordinates,
x̃1
′
(r) =
√∣∣∣∣3λ2
∣∣∣∣ 1ς
∫
dr
(
1 − M
r
− λr2
)−1/2
,
x̃2
′ = θ, y3′ = ϕ, y4′ = t;
g̊
1′ = 1,g̊2′ (̃x1
′
) = r2(̃x1′),
h̊3′ = r2(̃x1′) sin(x2′),
h̊4′ = −
(
1 − M
r (̃x1′)
+ λr2(̃x1′)
)
,
(99)
the metric (97) is written as a “prime” metric
ds2 =g̊
α′β ′ (̃x
k′)duα
′
duβ
′ =g̊
1′(dx̃
1′)2 +g̊
2′ (̃x
1
′
)(dx̃2
′
)2
+h̊3′ (̃x1
′
,x̃2
′
)(dy3
′
)2 +h̊4′ (̃x1
′
)(dy4
′
)2,
for some constants M , λ and uα =
(̃xk
′
, ya). In order to
work with a “formal” off-diagonal metric of type (75) with
nontrivial valuesh̊∗a,ẘi andn̊i , butW̊αβγ (̃uμ) = 0, see (4),
we consider a coordinate transform uα
′ = uα′(uα) with ϕ =
ϕ(y4,x̃ k) and t = t (y3,x̃ k), where
dt = ∂t
∂y3
[dy3 + (∂3t)−1(̃∂k t)d̃xk]and
dϕ = ∂ϕ
∂y4
[dy4 + (∂4ϕ)−1(̃∂kϕ)d̃xk]
for∂̃iϕ = ∂ϕ/∂x̃ i and ∂aϕ = ∂ϕ/∂ya . Choosing
n̊i =∂̃i n(xk) = (∂3t)−1(̃∂i t), and
ẘi =∂̃i̊ /̊∗ = (∂4ϕ)−1(̃∂iϕ),
we express (97) as
ds2 =g̊
1′(dx̃
1′)2 +g̊
2′ (̃x
1
′
)(dx̃2
′
)2 +g̊
3
(xk (̃xk
′
))
×[dy3 +n̊i (̃xk)dx̃ i ]2 +g̊4[dy4 +ẘi (̃xk)dxi ]2,
g̊
4
(̃xk (̃xk
′
)) = (∂4ϕ)2r2(̃x1′) sin(̃x2′) and
g̊
3
(̃xk (̃xk
′
))
= −(∂3t)2
(
1 − M
r
+ λr2
)
.
(100)
The prime d-metric (100) allows us to apply the AFDM and
construct ε-deformations of geometric/physical objects and
physical parameters as shown in Sect. 2.4.
Forg̊
3
=h̊3(̃x1′) = (1 − Mr + λr2) and (∂3t)2 = 1 and
anisotropically polarized massM̃(ϕ) = M[1+ε cos(ω0ϕ+
ϕ0)], we obtain
s=0h3 = −
(
1 − M
r
+ λr2
)[
1 − εM
r
cos(ω0ϕ + ϕ0)
1 − Mr + λr2
]
=h̊3(̃x1
′
)
[
1 − εM
r
(h̊3)
−1 cos(ω0ϕ + ϕ0)
]
 −
[
1 −M̃(ϕ)
r
+ λr2
]
.
The parametric equation of an ellipse with radial parameter
r̊+ = M and eccentricity ε, r+ 
M
1−ε cos(ω0ϕ+ϕ0) , can be
determined in a simple way for λ = 0. We have to find solu-
tions of a third order algebraic equation in order to determine
possible horizons for nontrivial λ.
We construct ellipsoidal deformations of d-metric (100)
if χ = ςχ = 8 Mr ς2̊−2 cos(ω0ϕ + ϕ0), with identifi-
cation (98). Following the same method as in the previous
subsection but for ςχ used for d-metric coefficients (83), we
compute
ς gi =g̊i [1 + εχi ] = [1 + εe
0q 1q/ 2ς2]g̊
i
solution of 2-d Poisson equations (39);
ςh3 = [1 + ε ςχ3]g̊3 =
[
1 − ε
1
8ς2g̊
3
̊2 ςχ
]
g̊
3
;
ςh4 = [1 + ε ςχ3]g̊3
=
[
1 + ε
(
2
(
ςχ+̊
̊∗
ςχ∗
)
+
1
8ς2g̊
4
̊2 ςχ
)]
g̊
3
;
ςni = [1 + ε nςχi ]n̊i =
[
1+εñi
∫
dy4
(
ςχ+̊
̊∗
ςχ∗
+ 5
16ς2
1
g̊
4
(̊2 ςχ)∗
)]
n̊i ,
ςwi = [1 + ε wς χi ]ẘi
=
[
1 + ε
(
∂i (
ςχ̊)
∂i̊
− (
ςχ̊)∗
̊∗
)]
ẘi ;
(101)
where
ñi (xk) is a redefined integration function including
contributions from the prime metric (100). The generating
functions ςχ and
0q can be determined for an ellipsoid
configuration induced by the effective cosmological constant
ς2 in R2 gravity.
Finally, we generate a class of generic off-diagonal met-
rics for ellipsoid Kerr–de Sitter configurations related to the
cosmological constant in heterotic string gravity,
ds2 = [1 + εe 0q 1q/ 2ς2][g̊
1′(dx̃
1′)2 +g̊
2′ (̃x
1
′
)(dx̃2
′
)2]
−
[
1 − ε
1
8ς2g̊
3
̊2 ςχ
]
g̊
3
(xk (̃xk
′
))
123
17 Page 26 of 32
Eur. Phys. J. C (2017) 77:17
×
[
dy3 +
[
1 + εñi
∫
dy4
(
ςχ +̊
̊∗
ςχ∗
+ 5
16ς2
1
g̊
4
(̊2 ςχ)∗
)]
n̊idx̃
i
]2
+
[
1 + ε
(
2
(
ςχ +̊
̊∗
ςχ∗
)
+
1
8ς2g̊
4
̊2 ςχ
)]
×g̊
4
[
dϕ +
[
1 + ε
(
∂i (
ςχ̊)
∂i̊
− (
ςχ̊)∗
̊∗
)]
ẘidx̃
i
]2
.
(102)
Such metrics have Killing symmetry on ∂/∂y3 and are
completely defined by generating functions 1q and ςχ and
effective source 2ς2 = K
induced from string theory.
They define ε-deformations of Kerr–de Sitter black holes
into ellipsoid configurations with effective (polarized) cos-
mological constants determined by constants in string theory
and equivalent MGTs. If the LC-conditions are satisfied, such
metrics can be modeled in GR with nontrivial cosmological
constant.
3.4 Extra-dimensional off-diagonal string modifications of
the Kerr solutions
Various classes of exact solutions in heterotic string gravity
can be constructed, which depend on which type of effective
sources (19) and (38) are chosen. As a result, various classes
of generic off-diagonal deformations of the Kerr metric into
higher-dimensional exact solutions can be constructed. In
this subsection we shall construct and analyze a series of 6-d
and 10-d solutions encoding possible higher-dimension inter-
actions with effective cosmological constants, warping con-
figurations, f -modifications and certain analogies to almost-
Kä hler gravity models.
3.4.1 6-d deformations with nontrivial cosmological
constant
Solutions are determined by certain configurations of the NS-
fieldsĤα1β1μ1 which are nontrivial on the first shell s = 1,
see Eqs. (21) and the respective term in (38),
Hϒμ1ν1 =
1
4
Ĥα1β1μ1Ĥ
α1β1
ν1
with effective constant H;
Hϒμ1ν1 = −
6
2
( Hs)2gβ1μ1 ,
for H = −3( Hs)2,
where the coefficient 3 is used for an effective 6-d space
time with trivial extension on four other internal space coor-
dinates. We introduce an effective source in the quadratic
element (96) extended with one nontrivial extra 2-d shell.
Such a family of generic off-diagonal stationary solutions is
described by
ds2Kη6d =
(
1 − e
0q 1q
12( Hs)2
)
[(dx1′)2 + (dx2′)2] + A[1
−2εζ̃ sin(ω0ϕ + ϕ0)]
[
dy3
′ +
(
ε∂k′
χn(xi
′
)
−∂k′
(
ŷ3
′ + ϕ B
A
))
dxk
′
]2
+
[
1 + ε
(
−[−24A (
Hs)2 +̊2 ]̃ζ
12( Hs)2̊2
sin(ω0ϕ + ϕ0)
+16̃ζω0A (
Hs)2
̊2
cos(ω0ϕ + ϕ0)
)]
×
(
C − B
2
A
)
[dϕ + ε∂i ′Ǎ)dxi
′ ]2
+
[
1 + ε
1̊2 H1 χ
12( Hs)2g̊
5
]
g̊
5
(xk (̃xk
′
))
×
[
dy5 +
[
1 + ε 1ñi1
∫
dy6
(
H
1 χ +
1̊
1̊∗1
H
1 χ
∗1
+
5
24( Hs)2g̊
6
( 1̊2 H1 χ)
∗1
)]
n̊i1 dx̃
i1
]2
+
[
1 + ε
(
2( H1 χ +
1̊
1̊∗1
H
1 χ
∗1)+
1̊
12( Hs)2g̊
6
H
1 χ
)]
g̊
6
×
[
dy6+
[
1+ε
(
∂i1(
H
1 χ
1̊)
∂i1
1̊
− (
H
1 χ
1̊)∗1
1̊∗1
)]
˜̊wi1 dx̃ i1
]2
.
(103)
In the above formulas, we consider a generating function
H
1 χ(̃x
i ′ , y4, y6), which results, in general, in nontrivial non-
holonomic torsions encoding contributions ofĤα1β1μ1 via
an effective cosmological constant ( Hs)2. One considers
a summation on i ′1 = 1, 2, 4 for terms like
∂i1 (
H
1 χ
1̊)
∂i1
1̊
˜̊wi1
dx̃ i1 .
Off-diagonal extra-dimensional gravitational interactions
modify a Kerr metric for any nontrivial cosmological con-
stant determined by a corresponding ansatz forĤα1β1μ1 in
6-d. In a similar form we can generalize the constructions in
8-d and 10-d with nontrivial extra shell componentsĤαsβsμs
and R = 0 in (19).
3.4.2 10-d deformations with NS 3-form and 6-d almost
Kähler internal spaces
The class of solutions (103) can be generalized for 10-d
spacetimes with indices of fields running over 10-d values
and nontrivial sources (34) redefined in (38), with nontriv-
123
Eur. Phys. J. C (2017) 77:17
Page 27 of 32
17
ial H = −5( Hs)2, F = −5nF ( Fs)2 and R =
−5trnR( Rs)2 in  = H+ F+ R. The corresponding
quadratic line element is
ds2Kη10d =
(
1 + e 0q
1q
4
)
[(dx1′ )2 + (dx2′ )2]
+A[1 − 2εζ̃ sin(ω0ϕ + ϕ0)]
×
[
dy3
′ +
(
ε∂k′
χn(xi
′
)− ∂k′
(
ŷ3
′ + ϕ B
A
))
dxk
′
]2
+
[
1 + ε
(
[8A+̊2 ]̃ζ
4̊2
sin(ω0ϕ + ϕ0)
− 4̃ζω0A 
̊2
cos(ω0ϕ + ϕ0)
)]
×
(
C − B
2
A
)
[dϕ + ε(∂i ′Ǎ)dxi ′ ]2
+
[
1 − ε 1
4g̊
5
1̊2 1 χ
]
g̊
5
(xk (̃xk
′
))
×
[
dy5 +
[
1 + ε 1ñi1
∫
dy6
(

1 χ +
1̊
1̊∗1

1 χ
∗1
− 5
8
1
g̊
6
( 1̊2 1 χ)
∗1
)]
n̊i1 dx̃
i1
]2
+
[
1 + ε
(
2
(

1 χ +
1̊
1̊∗1

1 χ
∗1
)
−
1̊
4g̊
6

1 χ
)]
g̊
6
×
[
dy6 +
[
1 + ε
(
∂i1 (

1 χ
1̊)
∂i1
1̊
− (

1 χ
1̊)∗1
1̊∗1
)]
˜̊wi1 dx̃ i1
]2
+
[
1 − ε 1
4g̊
7
2̊2 2 χ
]
g̊
7
(xk (̃xk
′
))
×
[
dy7 +
[
1 + ε 2ñi2
∫
dy8
(

2 χ +
2̊
2̊∗2

2 χ
∗2
− 5
8
1
g̊
8
( 2̊2 2 χ)
∗2
)]
n̊i2 dx̃
i2
]2
+
[
1 + ε
(
2
(

2 χ +
2̊
2̊∗2

2 χ
∗2
)
−
2̊
4g̊
8

2 χ
)]
g̊
8
×
[
dy8 +
[
1 + ε
(
∂i2 (

2 χ
2̊)
∂i2
2̊
− (

2 χ
2̊)∗2
2̊∗2
)]
˜̊wi2 dx̃ i2
]2
+
[
1 − ε 1
4g̊
9
3̊2 3 χ
]
g̊
9
(xk (̃xk
′
))[dy9 +
[
1 + ε 3ñi3
×
∫
dy10
(

3 χ+
3̊
3̊∗3

3 χ
∗3 − 5
8
1
g̊
10
( 3̊2 3 χ)
∗3
)]
n̊i3 dx̃
i3 ]2
+
[
1 + ε
(
2
(

3 χ +
3̊
3̊∗3

3 χ
∗3
)
−
3̊
4g̊
10

3 χ
)]
×g̊
10
[
dy10+
[
1+ε
(
∂i3 (

3 χ
3̊)
∂i3
3̊
− (

3 χ
3̊)∗3
3̊∗3
)]
˜̊wi3 dx̃ i3
]2
.
(104)
This generic off-diagonal stationary metric for 10-d space-
times defines a very general class of stationary solutions
of the nonholonomic equations of motion in heterotic grav-
ity (12)–(15) with effective scalar field encoded into the N-
connection structure. It is important to use “shell by shell”
nonholonomic variables for the quadratic element (104).
Only in such cases we can understand the nonlinear sym-
metries and classify the types of generating and integration
functions corresponding to horizontal and higher order verti-
cal conventional sources. Such properties cannot be encoded
in a minimal form if notations for indices running coordinate
values from 0 to 9 are used as in standard papers on heterotic
supergravity. For small ε-deformations, the 4-d component of
such metrics describes a Kerr type black ellipsoid with eccen-
tricity ε and ϕ-anisotropic polarization of physical constants
and horizons. The nonholonomic deformations also encode
sources from all gauge-like and interior space gravitational
fields via a redefinition of generating functions.
Similar classes of non-vacuum solutions can also be
modeled for Einstein–Finsler spaces if extra-dimensional
coordinates are treated as velocity/momentum coordinates
[10,24,26,43,44,53,54]. The metrics possess a respective
Killing symmetry on ∂t and ∂/∂y9. Using the AFDM, we
can construct solutions depending on all 10-d coordinates
which may describe geometric evolution and time propagat-
ing Kerr black holes determined by heterotic string gravity
effects. They define ε-deformations of Kerr–de Sitter black
holes into ellipsoid configurations with effective cosmolog-
ical constants determined by constants in GR, possible f -
modifications and extra-dimension contributions [72]. By
nonholonomic frame transforms and distortions of the canon-
ical d-connection, the above metric can be rewritten in canon-
ical almost-Kähler variables on 6-d internal space, as we
prove in [48]. We omit such constructions (being very impor-
tant in various models of deformation and brane quantization
of MGTs and geometric evolution theories [10,11,63,70]) in
this work.
3.4.3 Off-diagonal solutions in standard 10-d heterotic
string coordinates
We can re-write the solution (104) in standard variables used
in [1–7] with coordinates having prime Greek indices rede-
fined in spherical 4-d coordinates (99) and extra dimensions,
xμ = (x0 = x0′ = t, x1 = x1′ = r, x2 = x2′
= ϑ, x3= x3′ =ϕ, x4=u5′ =u5,..., x9= vu10′ =u10),
considering respective partial derivatives and explicit values
for the functions A, B,C (87) determined by the 4-d Kerr
black hole solution and explicit parameterizations of integra-
tion and generating functions,
123
17 Page 28 of 32
Eur. Phys. J. C (2017) 77:17
ds2Kη10d =
(
1 + e 0q(r)
1q(r)
4
)
[(dr)2 + ϑ2(dr)2]
− r
2 − 2m0 + a2 − a2 sin2 ϑ
r2 + a2 cos2 ϑ
[1 − 2εζ̃ sin(ω0ϕ + ϕ0)]
×
[
dt+
(
ε∂r
χn(r,ϑ)− ∂r
[
ŷ3
′
(r,ϑ,ϕ)+ ϕ
2m0a sin2 ϑ
(r2−2m0+a2−a2 sin2 ϑ)
])
dr
+
(
ε∂ϑ
χn(r,ϑ)− ∂ϑ
[
ŷ3
′
(r,ϑ,ϕ) +ϕ
2m0a sin2 ϑ
(r2 − 2m0 + a2 − a2 sin2 ϑ)
])
dϑ
]2
+
⎡
⎣1 + ε
⎛
⎝ [̊2(r,ϑ,ϕ)− 8A r
2−2m0+a2−a2 sin2 ϑ
r2+a2 cos2 ϑ
]̃ζ
4̊2(r,ϑ,ϕ)
sin(ω0ϕ + ϕ0)
− 4̃ζω0 A 
̊2
cos(ω0ϕ + ϕ0)
)](
sin2 ϑ
[
(r2 + a2)2 −a2 sin2 ϑ]
r2 + a2 cos2 ϑ
+
4(m0)2a2 sin4 ϑ
(r2 + a2 cos2 ϑ)(r2 − 2m0 + a2 − a2 sin2 ϑ)
)
×[dϕ + ε(∂rǍ(r,ϑ,ϕ))dr + ε(∂ϑǍ(r,ϑ,ϕ))dϑ]2
+
[
1 − ε
1
4g̊
5
(r,ϑ,ϕ)
1̊2(r,ϑ,ϕ, x5) 1 χ(r,ϑ,ϕ, x
5)
]
×g̊
4
(r,ϑ,ϕ)
[
dx4 +
[
1 + ε 1ñ1(r,ϑ,ϕ)
∫
dx5( 1 χ(r,ϑ,ϕ, x
5)
+
1̊(r,ϑ,ϕ, x5)
∂
∂x5
1̊(r,ϑ,ϕ, x5)
∂
∂x5

1 χ(r,ϑ,ϕ, x
5)
− 5
8
1
g̊
5
(r,ϑ,ϕ)
∂
∂x5
( 1̊2(r,ϑ,ϕ, x5) 1 χ(r,ϑ,ϕ, x
5)))
]
1n̊1(r,ϑ,ϕ)dr
+ε 1ñ2(r,ϑ,ϕ)
∫
dx5( 1 χ(r,ϑ,ϕ, x
5)
+
1̊(r,ϑ,ϕ, x5)
∂
∂x5
1̊(r,ϑ,ϕ, x5)
∂
∂x5

1 χ(r,ϑ,ϕ, x
5)− 5
8
1
g̊
5
(r,ϑ,ϕ)
× ∂
∂x5
( 1̊2(r,ϑ,ϕ, x5) 1 χ(r,ϑ,ϕ, x
5)))
]
1n̊2(r,ϑ,ϕ)dϑ
+ε 1ñ3(r,ϑ,ϕ)
∫
dx5
(

1 χ(r,ϑ,ϕ, x
5)
+
1̊(r,ϑ,ϕ, x5)
∂
∂x5
1̊(r,ϑ,ϕ, x5)
∂
∂x5

1 χ(r,ϑ,ϕ, x
5)− 5
8
1
g̊
5
(r,ϑ,ϕ)
∂
∂x5
× ( 1̊2(r,ϑ,ϕ, x5) 1 χ(r,ϑ,ϕ, x5))
)]
1n̊3(r,ϑ,ϕ)dϕ]2
+
[
1 + ε
(
2
(

1 χ(r,ϑ,ϕ, x
5) +
1̊(r,ϑ,ϕ, x5)
∂
∂x5
1̊(r,ϑ,ϕ, x5)
∂
∂x5

1 χ(r,ϑ,ϕ, x
5)
)
−
1̊(r,ϑ,ϕ, x5)
4g̊
5
(r,ϑ,ϕ)

1 χ(r,ϑ,ϕ, x
5)
)]
g̊
5
(r,ϑ,ϕ)
[
dx5
+
[
1 + ε
(
∂r (

1 χ(r,ϑ,ϕ, x
5) 1̊(r,ϑ,ϕ, x5))
∂r 1̊(r,ϑ,ϕ, x5)
−
∂
∂x5
( 1 χ(r,ϑ,ϕ, x
5) 1̊(r,ϑ,ϕ, x5))
∂
∂x5
1̊(r,ϑ,ϕ, x5)
)]
1˜̊w1(r,ϑ,ϕ)dr
+
[
1 + ε
(
∂ϑ (

1 χ(r,ϑ,ϕ, x
5) 1̊(r,ϑ,ϕ, x5))
∂ϑ 1̊(r,ϑ,ϕ, x5)
−
∂
∂x5
( 1 χ(r,ϑ,ϕ, x
5) 1̊(r,ϑ,ϕ, x5))
∂
∂x5
1̊(r,ϑ,ϕ, x5)
)]
1˜̊w2(r,ϑ,ϕ)dϑ
+
[
1 + ε
(
∂ϕ(

1 χ(r,ϑ,ϕ, x
5) 1̊(r,ϑ,ϕ, x5))
∂ϕ 1̊(r,ϑ,ϕ, x5)
−
∂
∂x5
( 1 χ(r,ϑ,ϕ, x
5) 1̊(r,ϑ,ϕ, x5))
∂
∂x5
1̊(r,ϑ,ϕ, x5)
)]
1˜̊w3(r,ϑ,ϕ)dϕ]2
+
[
1 − ε
1
4g̊
6
(r,ϑ,ϕ, x5)
2̊2(r,ϑ,ϕ, x5, x7) 2 χ(r,ϑ,ϕ, x
5, x7)
]
×g̊
6
(r,ϑ,ϕ, x5)
×
[
dx6 +
[
1 + ε 2ñr (r,ϑ,ϕ, x5)
∫
dx7
(

2 χ(r,ϑ,ϕ, x
5, x7)
+
2̊(r,ϑ,ϕ, x5, x7)
∂
∂x7
2̊(r,ϑ,ϕ, x5, x7)
∂
∂x7

2 χ(r,ϑ,ϕ, x
5, x7)
− 5
8
1
g̊
7
(r,ϑ,ϕ, x5)
∂
∂x7
( 2̊2(r,ϑ,ϕ, x5, x7) 2 χ
× (r,ϑ,ϕ, x5, x7))
)]
2n̊r (r,ϑ,ϕ, x
5)dr
+
[
1 + ε 2ñϑ (r,ϑ,ϕ, x5)
∫
dx7
(

2 χ(r,ϑ,ϕ, x
5, x7)
+
2̊(r,ϑ,ϕ, x5, x7)
∂
∂x7
2̊(r,ϑ,ϕ, x5, x7)
∂
∂x7

2 χ(r,ϑ,ϕ, x
5, x7)
− 5
8
1
g̊
7
(r,ϑ,ϕ, x5)
∂
∂x7
( 2̊2(r,ϑ,ϕ, x5, x7) 2 χ(r,ϑ,ϕ, x
5, x7))
)]
× 2n̊ϑ (r,ϑ,ϕ, x5)dϑ
+
[
1 + ε 2ñϕ(r,ϑ,ϕ, x5)
∫
dx7
(

2 χ(r,ϑ,ϕ, x
5, x7)
+
2̊(r,ϑ,ϕ, x5, x7)
∂
∂x7
2̊(r,ϑ,ϕ, x5, x7)
∂
∂x7

2 χ(r,ϑ,ϕ, x
5, x7)
− 5
8
1
g̊
7
(r,ϑ,ϕ, x5)
∂
∂x7
( 2̊2(r,ϑ,ϕ, x5, x7) 2 χ(r,ϑ,ϕ, x
5, x7))
)]
× 2n̊ϕ(r,ϑ,ϕ, x5)dϕ
+
[
1 + ε 2ñ5(r,ϑ,ϕ, x5)
∫
dx7
(

2 χ(r,ϑ,ϕ, x
5, x7)
+
2̊(r,ϑ,ϕ, x5, x7)
∂
∂x7
2̊(r,ϑ,ϕ, x5, x7)
∂
∂x7

2 χ(r,ϑ,ϕ, x
5, x7)
− 5
8
1
g̊
7
(r,ϑ,ϕ, x5)
∂
∂x7
( 2̊2(r,ϑ,ϕ, x5, x7) 2 χ(r,ϑ,ϕ, x
5, x7))
)]
× 2n̊5(r,ϑ,ϕ, x5)dx5
]2
+
[
1 + ε
(
2
(

2 χ(r,ϑ,ϕ, x
5, x7)+
2̊(r,ϑ,ϕ, x5, x7)
∂
∂x7
2̊(r,ϑ,ϕ, x5, x7)
× ∂
∂x7

2 χ(r,ϑ,ϕ, x
5, x7)
)
−
2̊(r,ϑ,ϕ, x5, x7)
4g̊
7
(r,ϑ,ϕ, x5, x7)

7 χ(r,ϑ,ϕ, x
5, x7)
)]
g̊
7
(r,ϑ,ϕ, x5)
×
[
dx7 +
[
1 + ε
(
∂r (

2 χ(r,ϑ,ϕ, x
5, x7) 2̊(r,ϑ,ϕ, x5, x7))
∂r 2̊(r,ϑ,ϕ, x5, x7)
−
∂
∂x7
( 2 χ(r,ϑ,ϕ, x
5, x7) 2̊(r,ϑ,ϕ, x5, x7))
∂
∂x7
2̊(r,ϑ,ϕ, x5, x7)
)]
2˜̊wr (r,ϑ,ϕ, x5)dr
+
[
1 + ε
(
∂ϑ (

2 χ(r,ϑ,ϕ, x
5, x7) 2̊(r,ϑ,ϕ, x5, x7))
∂ϑ 2̊(r,ϑ,ϕ, x5, x7)
−
∂
∂x7
( 2 χ(r,ϑ,ϕ, x
5, x7) 2̊(r,ϑ,ϕ, x5, x7))
∂
∂x7
2̊(r,ϑ,ϕ, x5, x7)
)]
× 2˜̊wϑ(r,ϑ,ϕ, x5)dϑ
+
[
1 + ε
(
∂ϕ(

2 χ(r,ϑ,ϕ, x
5, x7) 2̊(r,ϑ,ϕ, x5, x7))
∂ϕ 2̊(r,ϑ,ϕ, x5, x7)
−
∂
∂x7
( 2 χ(r,ϑ,ϕ, x
5, x7) 2̊(r,ϑ,ϕ, x5, x7))
∂
∂x7
2̊(r,ϑ,ϕ, x5, x7)
)]
× 2˜̊wϕ(r,ϑ,ϕ, x5)dϕ
+
[
1 + ε
(
∂
∂x5
( 2 χ(r,ϑ,ϕ, x
5, x7) 2̊(r,ϑ,ϕ, x5, x7))
∂
∂x5
2̊(r,ϑ,ϕ, x5, x7)
−
∂
∂x7
( 2 χ(r,ϑ,ϕ, x
5, x7) 2̊(r,ϑ,ϕ, x5, x7))
∂
∂x7
2̊(r,ϑ,ϕ, x5, x7)
)]
2˜̊w5
123
Eur. Phys. J. C (2017) 77:17
Page 29 of 32
17
× (r,ϑ,ϕ, x5)dx5
]2
+
[
1 − ε
1
4g̊
8
(r,ϑ,ϕ, x5, x6, x7)
3̊2
×(r,ϑ,ϕ, x5, x6, x7, x9) 3 χ(r,ϑ,ϕ, x5, x6, x7, x9)
]
g̊
8
(r,ϑ,ϕ, x5, x6, x7)
×
[
dx8 +
[
1 + ε 3ñr (r,ϑ,ϕ, x5, x6, x7)
∫
dx9
×( 3 χ(r,ϑ,ϕ, x5, x6, x7, x9)+
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9

3 χ
×(r,ϑ,ϕ, x5, x6, x7, x9)− 5
8
1
g̊
9
(r,ϑ,ϕ, x5, x6, x7)
∂
∂x9
×( 3̊2 (r,ϑ,ϕ, x5, x6, x7, x9) 3 χ
× (r,ϑ,ϕ, x5, x6, x7, x9))
)]
3n̊r (r,ϑ,ϕ, x
5, x6, x7)dr
+
[
1 + ε 3ñϑ (r,ϑ,ϕ, x5, x6, x7)
∫
dx9
(

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)
+
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)
− 5
8
1
g̊
9
(r,ϑ,ϕ, x5, x6, x7)
∂
∂x9
×( 3̊2(r,ϑ,ϕ, x5, x6, x7, x9) 3 χ(r,ϑ,ϕ, x5, x6, x7, x9))
)]
× 3n̊ϑ (r,ϑ,ϕ, x5, x6, x7)dϑ +
[
1 + ε 3ñϕ(r,ϑ,ϕ, x5, x6, x7)
×
∫
dx9
(

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)
+
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)
− 5
8
1
g̊
9
(r,ϑ,ϕ, x5, x6, x7)
∂
∂x9
( 3̊2 (r,ϑ,ϕ, x5, x6, x7, x9) 3
× χ(r,ϑ,ϕ, x5, x6, x7, x9))
)]
3n̊ϕ(r,ϑ,ϕ, x
5, x6, x7)dϕ
+
[
1 + ε 3ñ5(r,ϑ,ϕ, x5, x6, x7)
∫
dx9( 3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)
+
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)
− 5
8
1
g̊
9
(r,ϑ,ϕ, x5, x6, x7)
∂
∂x9
( 3̊2 (r,ϑ,ϕ, x5, x6, x7, x9) 3
×χ(r,ϑ,ϕ, x5, x6, x7, x9)))] 3n̊5(r,ϑ,ϕ, x5, x6, x7)dx5
+
[
1 + ε 3ñ6(r,ϑ,ϕ, x5, x6, x7)
∫
dx9
(

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)
+
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)
− 5
8
1
g̊
9
(r,ϑ,ϕ, x5, x6, x7)
∂
∂x9
( 3̊2 (r,ϑ,ϕ, x5, x6, x7, x9) 3
× χ(r,ϑ,ϕ, x5, x6, x7, x9))
)]
3n̊6
×(r,ϑ,ϕ, x5, x6, x7)dx6 +
[
1 + ε 3ñ7(r,ϑ,ϕ, x5, x6, x7)
×
∫
dx9
(

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)
+
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)
− 5
8
1
g̊
9
(r,ϑ,ϕ, x5, x6, x7)
∂
∂x9
( 3̊2 (r,ϑ,ϕ, x5, x6, x7, x9) 3
× χ(r,ϑ,ϕ, x5, x6, x7, x9))
)]
3n̊7
× (r,ϑ,ϕ, x5, x6, x7)dx7
]2 +
[
1 + ε
×
(
2
(

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)+
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
× ∂
∂x9

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9)
)
−
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
4g̊
9
(r,ϑ,ϕ, x5, x6, x7, x9)

3
× χ(r,ϑ,ϕ, x5, x6, x7, x9)
)]
g̊
9
(r,ϑ,ϕ, x5, x6, x7)
×
[
dx9 +
[
1 + ε
(
∂r (

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂i3
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
−
∂
∂x9
( 3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
)]
3
ט̊wr (r,ϑ,ϕ, x5, x6, x7)dr
+
[
1 + ε
(
∂r (

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂r 3̊(r,ϑ,ϕ, x5, x6, x7, x9)
−
∂
∂x9
( 3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
)]
3
ט̊wr (r,ϑ,ϕ, x5, x6, x7)dr
+
[
1 + ε
(
∂ϑ (

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂ϑ 3̊(r,ϑ,ϕ, x5, x6, x7, x9)
−
∂
∂x9
( 3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
)]
3
ט̊wϑ(r,ϑ,ϕ, x5, x6, x7)dϑ
+
[
1 + ε
(
∂ϕ(

3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂ϕ 3̊(r,ϑ,ϕ, x5, x6, x7, x9)
−
∂
∂x9
( 3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
)]
3
ט̊wϕ(r,ϑ,ϕ, x5, x6, x7)dϕ
+
[
1 + ε
(
∂
∂x5
( 3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂
∂x5
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
−
∂
∂x9
( 3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
)]
3
ט̊w5(r,ϑ,ϕ, x5, x6, x7)dx5
+
[
1 + ε
(
∂
∂x6
( 3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂
∂x6
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
−
∂
∂x9
( 3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
)]
3
ט̊w6(r,ϑ,ϕ, x5, x6, x7)dx6
+
[
1 + ε
(
∂
∂x7
( 3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂
∂x7
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
−
∂
∂x9
( 3 χ(r,ϑ,ϕ, x
5, x6, x7, x9) 3̊(r,ϑ,ϕ, x5, x6, x7, x9))
∂
∂x9
3̊(r,ϑ,ϕ, x5, x6, x7, x9)
)]
× 3˜̊w7(r,ϑ,ϕ, x5, x6, x7)dx6]2.
(105)
The above quadratic elements transform into a trivial embed-
ding of the 4-d Kerr solution into a 10-d spacetime if ε → 0
and primary data (labeled by circles) can be transformed into
a diagonal 6-d internal space.
The solution (105) is equivalent to (104) up to a redefi-
nition of the coordinates and some integration functions. It
is difficult to see “shell by shell” nonlinear symmetries and
construct generic off-diagonal solutions in the variant with
standard variables used in heterotic string gravity. In for-
mer non N-adapted variables, the formulas are much more
cumbersome and less adapted for studying the limits to well-
known solutions and generalizations with nontrivial back-
123
17 Page 30 of 32
Eur. Phys. J. C (2017) 77:17
grounds and extra-dimensional contributions. To elaborate
and apply a corresponding “shell by shell” N-adapted geo-
metric techniques of constructing exact solutions in 4d and
extra-dimension theories is important both from a mathemat-
ical stand point and a physical point of view.
4 Outlook and concluding remarks
In this work, we have applied the anholonomic frame method,
AFDM, for constructing new classes of stationary solu-
tions of motion equations in heterotic supergravity. Such
solutions have generic off-diagonal metrics for effective
ten-dimensional, 10-d, spacetimes enabled with general-
ized connections and depend on all possible 4-d and extra-
dimensional space coordinates. They admit subclasses of
solutions with warping on coordinate y4, nearly almost-
Kähler 6-d internal manifolds in the presence of nonholo-
nomically deformed gravitational and gauge instantons. The
almost-K ähler structure is necessary if we want to gener-
ate the Kerr metric with possible (off-) diagonal and non-
holonomic deformations to black ellipsoid type solutions
with locally anisotropic polarized physical constants, small
deformations of horizons, embedding into nontrivial extra-
dimension vacuum gravitational fields and/or gauge configu-
rations. These solutions preserve two real supercharges cor-
responding to N = 1/2 supersymmetry from the viewpoint
of four non-compact dimensions and various nonholonomic
deformations.
Following the AFDM, we can integrate the equations of
motion in heterotic supergravity in very general forms with
dependence on all 10-d spacetime coordinates. Such con-
structions are possible to more general classes of almost-
Kähler nonholonomic variables and so-called canonical non-
holonomic variables. This allows us to decouple string modi-
fied Einstein equations with effective sources in general form
which are similar to Einstein–Yang–Mills–Higgs, EYMH,
systems in higher-dimensions and with generalized gauge-
like interactions. It is possible to consider associated SU (3)
structures and solve generalized BPS equations and Bianchi
identities. A crucial difference from former approaches is
that our geometric methods allow us to work with gener-
ating and integration functions for off-diagonal metrics and
connections transforming equations of motion into nonlinear
systems of partial differential equations, PDEs. In particular,
we can reproduce former results for a diagonalizable ansatz
with dependence on radial and warping coordinates as solu-
tions of ordinary differential equations, ODEs.
To illustrate the power and importance of the AFDM as the
most general geometric method of constructing analytic solu-
tions of (modified) motion/gravitational and field equations,
we show how this formalism can be applied for generating N-
adapted (i.e. adapted to nonlinear connection structures) YM
and instanton configurations with possible associated SU (3)
nonholonomic structures; see the associated work [48]. New
classes of exact solutions describing small parametric modi-
fications of Kerr metrics with effective string sources are pro-
vided. We show that in a certain sense, a large class of phys-
ical effects in modified gravity models like R2 can be equiv-
alently modeled/explained by nonholonomic constraints and
effective sources in heterotic string gravity. In explicit form,
exact/parameteric extra dimension deformations of the black
hole metrics in 6-d and 10-d gravity with NS-3 form and 6-d
almost-Kähler internal spaces are constructed and analysed.
Finally, we emphasize that there are a plethora of future
directions which can be pursued using our methods and
results as starting points. This includes the construction of
cosmological solutions in the heterotic string gravity and/or
the study of smooth compact nonholonomic varieties in both
heterotic and geometric flow context. Similar analyses can be
performed in type II string theory in particular, including the
Ramond–Ramond sector and/or considering geometric flows
on internal spaces. We worked to the lowest order of α′ but
possibilities in the AFDM exist to extend the constructions
to higher orders.
Acknowledgements The SV research is for the QGR-Topanga with
a former partial support by IDEI, PN-II-ID-PCE-2011-3-0256, and
DAAD. He is grateful to N. Mavromatos, D. Lüst, O. Lechtenfeld, S. D.
Odintsov and C. Castro Perelman for valuable discussions and support.
This work contains also a summary of results of a talk at GR21 in NY.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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